I would say...
"Bournemouth Approach, G-ABCD, request zone transit"
"G-ABCD, Pass your message"
"G-ABCD is a PA-28 from ... to ..., currently at Romsey altitude 1800 feet on QNH 1010, request transit from Stoney Cross to Sandbanks via your overhead" (or whatever route I wanted)
They would say either yes (G-ABCD you are cleared to transit ..... squwark 1234, not above 2000 feet on .... ) or no (G-ABCD, negative ... remain clear of controlled airspace)