Hmmmmmm , 10 amp is kinda big for a 2200 watt generator .
You could try it .
120v is easy , use a good plug
If your wires are black and white you hook the black wire to the gold screw white wire to the silver screw .
Pump wires are yellow black red and green .
what type of pump?
I think the problem will be start surge most pumps will draw 3 or 4 times the power to start
so a 10amp pump could draw 30 amps a stall the generator .
My buddy is a well guy and he uses a 5500watt generator to start the pump .
For just running that 10 amps times 120 volts equals 1200 watts. but like has been said starting draw can be triple that, Same with older air conditioners. It might make a difference if your well is 20 feet or 200 feet dep tho. I have worked with a gas engine powered air compressor that wouldn't start till the tank was drained then it would start from zero air pressure and build up to full pressure...
for intermittent use maybe a cheap 4500 genny? I have a backup 12 volt well pump that is rated to pump from 200 ft deep.
I have a male / female plug installed within my well pump shed. I have a custom made 50’ extension cord (proper gauge). During power outages (hurricanes) I unplug the power lead at the pump motor and plug it into the extension cord and into my generator. The extension is pretty heavy so I store it on a wheeled garden hose cart. I keep my generator inside my privacy fenced yard So I need the long extension cord to reach it.
My well pump motor is 1hp 230 volt and my 4,500 watt generator grunts when the motor starts but is fine when the pump is running steady.