97 Corolla won't start. New battery, spark plugs and HT wires. Carburetor coughing smoke

The shape of a carburetor creates low pressure inside, fuel jets spray fuel into the chamber where it is mixed thoroughly and pulled into the engine's cylinders. The fuel to air mix needs to be a certain ratio for the engine to run, too much fuel to air means it is too rich, too little fuel means it is too lean. Putting your hand on the air intake restricts the flow of air, increasing the fuel to air ratio, if your car starts when you do this then you could have one of a few problems:

1. Dirty carb jets: if the jets get clogged up, and they always do eventually, then there won't be enough fuel supply and the engine won't run. This is especially true when the engine is cold. Sometimes this sort of problem can be fixed by spraying in carburetor cleaner and running a can of seafoam through the gas tank, if that doesn't work the carb may need a rebuild

2. Faulty choke: An engine needs a higher fuel to air ratio when it is cold, most carbs have a system which closes a flap when cold to restrict airflow, which then opens as the engine warms. If this breaks then you'll have trouble starting the engine when it is cold, but then it will run fine when heated up

3. Broken carburetor: there are mechanical parts which could have broken inside the carburetor, in this case you would need a repair/rebuild or a replacement carb

4. Vacuum leaks: engines rely heavily on suction to operate, if there is a leak in the vacuum system you can have hard starts and rough running. One trick is to spray carb cleaner at the base of the carburetor when the engine is running rough, if it smooths out temporarily it could be the carb is loose

One note: be careful with carb cleaner, it's extremely flammable!

I would invest in a can of carburetor cleaner and a can of seafoam first, see if you can clean things out and get it working again without major investment. Failing that it is digging into the carburetor and vacuum system.