what could cause the car to turn over but have a hard time starting?

Start by check fuel system.

1 hour (or less) of diagnostics by a competent shop/mechanic/tech/whateverthehellyuowanttocallem might save A LOT of MONEY and guesswork! Just a thought , next time it's in the situation you describe (hard re-start when warm) , try putting the gas pedal ALL the way to the floor and HOLDING it there , while cranking and see if it starts easier. Bring that info with you.