Hey. Is this happening only with War Thunder? If so,it is related to the game.
First thing to try,take the batteries out of the controller and hold the power button for 10 seconds,then replace both batteries. See if this works.
Quit the game(right pause button then quit on game from home screen to close app) or take out the disk. Now,you could try restarting the Xbox one(not sleep mode then back on but actually restart it). Although,It’s best to hold the power button on the Xbox itself and wait for it to turn off. This does a full power off. Then unplug the power supply from the wall,and hold the power button on the console to drain the charging brick(about 10 seconds to be safe). Then unplug the brick from the Xbox,then press the power button for another 10 seconds. Now plug it all back in and power it back on. Try it again.
You can also hook the remote up by the wire to the console. Then check if there’s a firmware update available for the controller. See if this fixes the issue.
If none of the options above fix the issue(being it’s not specific to the one game),it’s going to be a bad controller most likely. You can always blow out the joystick with canned air I guess. But I doubt this would fix it in this scenario. Your last option is to take it apart and you can check the little white wheels(2 of them) attached to the joystick. These are variable resistors that allow more electricity to pass depending on how much contact they are making with the circuit. When you move the joystick,it moves these little wheels. If one of them is dirty somehow,it could cause a weird issue. But this would be extremely difficult for someone inexperienced.