Q: Hesitation before engine starts up.

Hello, do you mean a lag "while" the starter turns the motor over, before it actually starts? Or a lag between turning the key to ’"start" and when the starter kicks off? A lag while the starter is turning, but the engine doesn’t fire could be a fuel system check valve that is leaking down while the car sits between use. A lag between when the ignition key is turned to "Start" and when the starter kicks the engine over could be caused by a corroded or poorly connected high current connection to the starter - say a battery cable connection. The corrosion introduces resistance that current literally has to burn through in order to pass enough current to spin the starter motor. It could possibly be a weak starter if there is not voltage drop from the batter to the starter itself. Not the alternator though.

I would recommend having the car’s starting issue inspected by a mobile, professional mechanic, such as one from YourMechanic, to diagnose this problem, get an accurate assessment of damage and cost estimate for repairs.