The Dome's no home for him! (3)
I believe the answer is:
ted
'him' is the definition.
'the dome's no home' is the wordplay.
I cannot quite see how this works, but
'the' could be 't' (the is pronounced as a 't' sound in some dialects) and 't' is found in the answer.
The remaining letters 'ed' is a valid word which might be clued in a way I don't see.
This explanation may well be incorrect...
'for' acts as a link.
Can you help me to learn more?
(Other definitions for ted that I've seen before include "Edward (dim.)" , "Heath or Hughes?" , "used to make hay" , "Edward, in short" , "Comic TV Father" .)