Are you sure it's a Hawk?
Does it say Hawk II on top?
The two barrel system was certainly Hawk mk II.
The Hawk that I had had a sloping end block ( like an Airsporter ). Maybe that slope was on the mark III as mine had a fixed barrel. The Vulcan and Excell had the square end like that looks to have?
The Hawks had a piston with twin rings ( I think Phospher Bronze ). These tended to bed into the main cylinder wall and were worn to different depths around their circumference. So if you took the piston out you had to try and make sure that the rings remained in the same place or they would not seal the same when re assembled.
The other seal is a standard Webley style seal at the breech that sits in the main cylinder.
It's the same as most things, but probably more with an old gun like this, it's worth what someone is willing to pay.
As a useable gun a low powered ( by today's standard ) gun like this will be worth very little so it's collectors, restorers that you are looking at.
I'm not sure the 0.22's consistently made 10fp. I think 8.5fp was probably more realistic with a standard spring. People may have got them up to 10fp but risked elongating the end plug hole.
I'd ask in the collector's section for value.
I'm afraid with it in need of a reblue and a stock refinish it's going to be well under 100 quid.