Wiring a hot tub pump
Better yet, you could replace the wire entirely and avoid splices but splicing it is an option as long as you have a junction box to make the joints in. Wire nutting and taping the conductors is not acceptable to the Code.

14 AWG wire is rated for 15 amp circuits so it has the ampacity to supply your hot tub pump. The heater on the hot tub is going to have to be on a separate circuit from the pump.