The starter engages and starts the motor fine when the motor is cold. however if you ride and try to restart the starter just spins and you can only start using the kick stater.

The two gears you have can be put on in reverse but are slightly different try swapping them and try it again

It acts like you have burnt spots on the starter shaft that touches the brushes ( voltage) to start. When you rock the vehicle in gear the starter shaft moves and it is placed in a position where the brushes can make a clean contact. Then when you turn the key, the starter will engage and the vehicle will start. My recommendation would be to change out the starter.

What clicks is the solenoid for the starter. I pushes the starter gear into the flywheel, then the starter motor is supposed to spin the flywheel and it starts. If you hear the click, but nothing else happens, there is a good chance that it is the starter itself. Check the wires going to the starter and make sure that the red one is 12 volts. The black one should go to ground. If it has 12V on it, then it is the starter motor. If it DOESN'T have 12V, then the problem is that power is not getting to the motor.

check that the pinion of the starter is engaging the flywheel teeth
check that the over run clutch of the pinion is engaging and not over running when it is supposed to lock up to turn the flywheel

Check to see that the pinion gear of the starter is hooting out and engaging the flywheel.
And if it is working it is possible that you have missing tooth/teeth on your flywheel and the starter cannot turn over the engine because of that.

Disconnect the battery and then carefully remove the starter. Take the unit to a local auto parts dealer for a bench test. Starters have two separate functions, one spins the shaft to crank the engine, the other is called a bendix, which pushes the starter gear out to engage the flywheel, and then after the engine engages, retracts it (except in the case of starter/generator combo units which recharge your battery and power your vehicle's electricals while the engine is running) Always best to get a bench test rather than depending on apparent symptoms.

Try taking a look at the starter and flywheel while attempting to start the engine. Does the small gear at the end of the starter move forward and engage the flywheel when you try to start the engine? Could it be stuck? The small gear must move forward to engage the flywheel. Keep your fingers away from the moving parts.

wore out not engaging, remove the cover and alighn the gears, make sure starter wire isn't bare youlll get shocked some times you have to cycle the starter a few times and help it alighn

sounds like the starter clutch is bad. the rollers get worn or the sprag gear gets worn so it will not grab and start the motor. check that out. you will have to take the engine side cover off to see it.

The starter has to throw it's gear out using centrifugal force. The speed of the starter has to be enough to overcome the spring inside. It is either a little seized and needs to be rebuilt and greased up, or the battery isn't powered up enough to get the right speed to engage properly. In that case clean the battery posts and clamps well, then charge or replace the battery. To know if the battery is bad wait till it is charged... if it still doesn't engage try again with booster cables from your car etc. and see if it suddenly works fine.