There is 1 wire. I ran a jumper from battery to the solenoid and it started right up. I'm wondering if there is a fuse link on the wire to the solenoid.How many wires on your solenoid. Usually there are two so you will have to run two wires. Don't anything about your specific solenoid but give it a try. It can't hurt anything.
I would first check the Seat Switch & the the Parking Brake Switch...
On the NH TC series the seat switch and the parking brake switch will not stop the engine from turning over however they cause the Fuel Solenoid to shut off. This will allow the engine to crank but not start.
You should also check to see if the Fuel Solenoid is activating when the key is turned run position. The fuel solenoid is on the right side of the tractor. If you set the parking brake and turn the Key to the ON position, you should hear it click.
IOW, it's filthy. Hope you take better care of the tractor than what the picture shows.If it were me I'd pull out 1 relay at a time and clean the sockets with contact cleaner . And reseat ur relays to start. lots of dirt there that can give you a bad connection.
Look at the color of the wire. Can you trace it back to its source? Go to the dealer and ask the Service Manager to provide you with a wiring diagram. Use the diagram to trace the wire back to all locations to the switch. ‘There is 1 wire. I ran a jumper from battery to the solenoid and it started right up. I'm wondering if there is a fuse link on the wire to the solenoid.