Martin: I have not mentioned a bent jet or needle. He has a HIF carb - internal jet and spring-loaded swinging needle. I expect you are thinking of a HS type where it is possible to damage the jet.

My gut feeling is that he is using the choke improperly. Even up here in cold Ottawa Canada a choke should only be needed on a cold start or if it has sat for several hours in winter weather. Not the case in California. His throttle and choke linkages may also be sticking. 

Great advice Dan, even a good reminder to us old-timers.
I was thinking that if the carb's jet was a little bent and was sticking a previous owner might have tried other adjustments without addressing a sticking jet.
Your comment about making sure the jet is free is very valid, one of the first things that I check.
Regards, Martin.

The issues you reported do not warrant a cleaning of the carb. You say it runs "perfectly fine" after a cold start, using the choke. That tells us the carb is in good condition. Taking it apart will get you into a rats-nest of other problems. You cannot "clean" it without a rebuild kit. For instance the seal for the fuel bowl will not go back on and you will need an assortment of gaskets. SU carbs generally stay pretty clean inside in use. The pictures I posted are of a used one I have.

Start with the simple things.
You live in mild California. You should only need a bit of choke on a cold start to "prime" the engine. Once it is running and not even fully warmed up, you should be able to run it with no choke. If it is sluggish you could add a bit of choke only until it smooths out. Once warm, no choke at all.

Once warmed, you should not need any choke at all. Or any throttle application.
NOTE: Don't waste your time pumping the accelerator - the SU carb DOES NOT HAVE an accelerator pump.

Your note about the choke not affecting rpms on the warm start suggests the linkage on the outside of the carb needs checking. Start with the throttle mechanism. It should move freely from the idle position to full open and close with just its spring. You should be able to spot the idle speed screw, though you should not need to adjust it. 

Check the choke mechanism is working like the throttle mechanism. Is the choke cable binding or stiff?. Do not disconnect it - there is a tiny part that holds the cable core that very quickly disappears down the back of the engine, never to be seen again.
Have a close look at the cam on the choke lever. As the choke turns, the cam comes in contact with an adjustment screw on the bottom of the throttle lever. Tat is the fast-idle adjustment screw.There may be a small arrow or mark on the choke cam too tell you where the fast-idle screw should first make contact. When set properly, you should be able to pull out a small bit of choke before the fast- idle starts to take effect.

Clean your air filter. It looks to be a permanent type that requires regular cleaning and re-oiling.
Check and clean your spark plugs and gap them to spec.
Check your spark plug wires are in good condition.
Check the timing.

You should be able to find the tuning specs for your engine in the Haynes manual - you did buy one, didn't you? It is the FIRST thing any new MIni owner needs to buy.

It looks like the oil level is low, I just ordered some more. I’ll let you know if that hanged anything. I’m also thinking about pulling off the carb and just giving it a quick clean to see if that helps.

Yogicoop:

Good photos!

You have a SU (brand name) HIF carburetor, with an aftermarket air cleaner. The "H" refers to horizontal flow (I think) and the "IF" refers to it having an integral fuel bowl. The fuel bowl is at the bottom of the carb.

The large chamber with the black cap is the "dashpot". When you remove the black cap, you will see a thin rod hanging from it with a brass lump at the end. If you look down the hole, you will see a smaller diameter bore that the brass lump fits into. That bore should have carb oil or motor oil in it. The brass lump and oil work as a "damper" or miniature shock absorber. The oil level should be just below the top of the inner bore - the brass lump will push out excess (don't worry about overflow - the engine will burn it.)

Here basically is how a SU carb works.
In the bottom off the carb is a long fuel jet with a specific inside diameter. The top of this jet is very close to the throat of the carb where the air passes into the engine. The air rushing by at varying speeds, it sucks fuel from the top of the jet. The fuel flow is regulated by a long thin tapered needle that slides out of the jet as the engine speed increases to allow more fuel in proportion to the amount of air flow. The air flow is regulated by the throttle (your right foot). So far so good?

The dashpot is the device that pulls the needle out as the engine goes faster. It works on vacuum generated by the flow of in-rushing air. The faster the air flows, the more vacuum and the higher the dashpot pulls the needle out. The dashpot has a big but gentle spring to push it back down and the vacuum works against the spring. (See picture of HIF and its spring.)

Because of the spring vs vacuum action, the dashpot core tends to bounce resulting in uneven fuel control. That is where the damper comes into play. It keeps the dashpot core from bouncing.

I have also posted 3 views of a HIF44. As you can see, the bottom is sealed.

One side view is the control side where the throttle and choke cables are connected. The "choke" on these carbs is not really a choke but a valve that lets additional fuel in through a separate passage. When you pull the choke cable, the mechanism opens the valve and a cam pushes the throttle open a little to give you a faster idle. When you push it back in, the cam should allow the throttle to settle back on its idle mixture screw. Yours may be sticking here somewhere.

On the other side view you can see several openings. The biggest one on the right is where it bolts to the intake manifold and you can see the throttle plate inside. Above it is a tiny, pointed port that is a vacuum connection for a distributor with vacuum advance. (Your car may not use it). It should either be connected to the distributor or capped or it poses a small air leak.

The next port to the left is the silvery one. It is the connection for crankcase ventilation. (Again your car may not use it). It should be either connected or capped. Otherwise it poses a big air leak.

The next one is the upper brass one. This is the fuel bowl air vent and needs to be open. However it very much should be connected to either a fuel vapour control system or at the very least a tube leading away from the engine and down to the air stream below the car. If the fuel bowl were to overflow, it would come out here (as well as flood the engine).

The lower brass tube is the fuel line connection. The fuel pump supplies fuel and just inside is a float valve to limit the amount of fuel the carb bowl should hold.

I'm not going to tell you where the fuel mixture screw is because touching it at this point will surely cause more problems!

Asking questions is good! Getting a service manual - even better! Our host sells it here,

https://www.minimania.com/part/646/Austin-Mini-Haynes-Workshop-Manual-1969-2001

EDIT:
From your pictures, the carb appears to have all the proper connections. Follow the hoses etc, to be sure there are no loose or cracked connections,

That thing on top of engine is your SU carburetor. Unscrew the black knob and see if you have oil in it. It should come up to where the threads stop. If you can remove the air cleaner, there will be a hole, stick your finger in the hole and gently see if the piston  will move freely up.

Dan,

I posted a couple of photos to assist my ignorance.  Is the fuel bowl not the thing on top of the carb? 

Okay next question:

What type of carb do you have?
SU HIF (the fuel bowl is on the bottom of the carb)?

SU HS (the fuel bowl is on the side of the carb with a small flexible tube going to the bottom of the main jet)?

Size: 38mm (1.5") or 44mm (1.75")?

Something else?

If the engine is tuned properly, you should only need the choke when the engine is cold. If the engine is warmed up, it should start without the choke.




Size?

I use the choke on both starts, when I move the choke on the second start there isn’t much change in the RPMs.

sounds like carb issue. do you choke it on initial start? then no choke on restart? What happens when you give it a little choke when running?

It’s has a 1275 carburetor engine

Welcome Yogicoop.

Does your engine have fuel injection of a carburetor? (Please verify).

I think a tune up kit might be warranted if it started poorly from cold as well as warmed up. Same thing for spark plugs.
A vacuum issue would probably present itself cold or warm too. However if it was caused by an improperly secure connection, expansion of parts (e.g the bolts holding the parts together) might allow a small leak to open up.

I’m looking for some advice. I just brought my 1991 Mini back with me from Japan. The issue I’m having is after the first start of the day the engine will turn on but whenever I press the accelerator the engine starts to stall. It runs perfectly fine the first time I start it, the issue only occurs after I shut it down and start it back up. Ive read it could be a vacuum issue and Ive also gotten advice to purchase a tune up kit and replace the spark plugs. I’m pretty new to older cars and would love to learn how to fix this issue myself before trying to find someone who works on older British cars in Northern California.  I appreciate any help I can get.