Turnigy servo motor needs push to start
Most likely the brushes have worn down and are intermittently shorting out the commutator. This is a very common failure mode in servos which use 'precious metal' brushed motors, especially if the servo is being worked hard. The usual way of dealing with this issue is to replace the servo well before reaching the end of its lifespan, and keep a spare on hand in case it fails earlier.

Repair would involve replacing the motor with an equivalent unit. Without specs getting the right motor may be tricky, and the pinion gear might be hard to move onto a new motor without damaging it. For a temporary repair you may be able to get a suitable motor from another servo of similar size but different model, or (if you are desperate) perhaps from a small motorized appliance. If you have another motor with identical brush assembly you could try exchanging it, but the brushes are very delicate and easy to damage.

For the future, replace servos on a regular preventative maintenance schedule, and don't buy from dodgy Chinese manufacturers!

Here's an example 'precious metal' brush assembly from a disassembled Mabuchi RF-020 6VDC motor. One of the brush fingers is already bent out of line.