I've had this piece on my mind for weeks. I'm certain that I've heard it lots of times on Classic FM or Radio 3, but haven't found anyone who can identify it. Very probably as soon as someone does name it I'll go 'duh!'.
I've created it as a Noteworthy file and saved as audio, with clarinet as the main tune and bassoon as an approximation to a bass harmony (it does something like that).
The MP3 file is
here.
(It feels as though it might be an Ave Maria, but it's not any of the obvious ones.)
ETA: Found!
G.F. Händel - Lascia ch'io pianga Via
Musipedia, which I hadn't known before, thanks
were_gopher.
Which illustrates the
point made by
osewalrus recently, that you don't have to know everything, knowing where to find it is just as good (and often more useful, as the referenced information may be more up to date or contain further links).