Where is the item in the cockpit I need?
Oh here it is. So after raising gear you need to left click and hold this hand pump lever for a while.
No, for normal operation you want to use the electrical switch on top of the control stick to activate the hydraulics to raise/lower the landing gear and flaps. As such, it helps vastly to assign that function to a switch on your joystick, since in reality you wouldn't be looking down into the cockpit searching for that switch, you'd simply move your thumb up to the switch and press it in (after selecting gear or flaps up/down). This way too it keeps your right hand on the control stick the whole time, as in reality, without having to fiddle with the mouse.
In the P40 Input Configurator, that function is named "Hydraulic Pump Switch". I use a CH Products Fighterstick, and I have the function assigned to a thumb button at the top of the stick, in much the same position as on the control stick of the real aircraft.
The big manual lever on the right floor of the cockpit is in the event that there are no electrics, and is used in normal operation only as a final assurance check to make sure the gear is fully locked up/down after the indicators show them as such (if the lever doesn't move, you know they're locked).
As Jeff Ethell explains, this is all leftover 1930's technology. Some other aircraft of its generation, like most original AT-6's, had a similar system.
