Cat given stroke the wrong way by Everyman sore at each end (7)

'cat' is the definition.
(I've seen this before)

'stroke the wrong way by everyman sore at each end' is the wordplay.
'stroke' becomes 'rub' (I've seen this in another clue).
'the wrong way' says the letters should be written backwards.
'by' says to put letters next to each other.
'everyman' becomes 'me' (where the crossword setter is 'Everyman').
'at each end' says to hollow out the word (remove centre letters) (take the end letters).
'sore' with its centre taken out is 'se'.
'rub' reversed gives 'bur'.
'bur'+'me'+'se'='BURMESE'

'given' acts as a link.