Starting issues when warm starts cold no problem

Take your battery to O'Reilly's or AutoZone and have the CCA (Cold Cranking Amps) checked.. it takes amperage to crank your ATV..not voltage..
DO NOT EVER JUMP YOUR ATV WITH A CAR OR HI-AMP BATTERY CHARGER..TOO MANY AMPS!!!!

Have a qualified professional clean and set the carburetor...it will make sure the starting process is easier...new spark plug too... and check the air filter...replace if not perfect!!!

The carburetor choke valve might be stuck open or not closing when the engine is cold. The engine needs a richer mixture to start. Too rich a mixture (flooding) can also cause trouble. Turn the fuel valve off to be sure the carb isn't leaking when not in use. If it smokes black on startup, it's flooding. Also check the condition of the spark plugs.

Is your choke working OK? Even in the summer time a cold engine needs a littel choke...more gas than air to get started. But you don't need to keep the choke on for very long in the summer since the engine warms up faster. You could try some carb cleaner or even taken the carb off and cleaning it good because it sounds definitely like a fuel problem so it was a good idea to check the pump and filter.

use the choke

Is it actually the warming up of the relays that helps it start though? My first guess would be the electrical contacts in the power distribution box. If starting is the only electrical issue you have, it's either a contact, the relay, or wiring to the PDB.

Try checking valve clearance. I had a Honda that did same thing, Set the valves and would start right up when cold.

start relay or just a failing silanoid my car did same i bypassed with a start switch click

As in...? Cold start as in warm weather not being run in a while, or cold weather every time it cools down? Is that fuel injected or carbureted?

clogged choke jet or passage for sure.

check battery u may need a jump or the oil is too cold and 4 wheeler needs to sit in a warm place for a hr or 2