Cable as in a wire moving through a sheath? The sheath needs to be stationary at both ends. Usually at one end, it is crimped in, usually the handle. The other end is held by a clamp over the sheath. If it was working before the cable broke and you have the correct cable, then it should just be a matter of adjusting the sheath away from or to the lever on the mower. If the wire inside the sheath pull to engage the deck, loosen the clamp and pull the sheath away a bit from the end of the wire. If the wire pushes to engage the mower loosen the clamp and push more sheath through the clamp to the end of the wire. It may take a few attempts to get it just right. Once you look at it and think about the way it works, you'll figure it out.

You will need to check for 12V and ground at the harness for starters. AN increased gap will not always let a clutch engage, but would it pop the fuse getting hot enough? Never say never. Is it possible a wire backed up in the connector behind the pto switch?

Does this have an electric PTO? If so there might be a wire that is getting hot and not allowing you to turn on the deck.

First question does everything work fine after the mower has cooled off? Could be that the starter is getting hot and not wanting to start. I don't know on that mower but if the deck engagement is electrical could be that the coil that engages the deck is also getting hot. I have an old Ford LGT that does the same thing.