Unfinished shelter causes offence to religious type up for the game (6)
I believe the answer is:
tennis
'game' is the definition.
(tennis is a kind of game)
'unfinished shelter causes offence to religious type up for the' is the wordplay.
'unfinished shelter causes' becomes 'en' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'offence to religious type' becomes 'sin'.
'up' shows that the letters should be reversed in order (in down clue: letters go upwards).
'for' means one lot of letters go next to another (I've seen this in other clues).
'the' becomes 't' (the is pronounced as a 't' sound in some dialects).
'sin' back-to-front is 'nis'.
'en'+'nis'='ennis'
'ennis' after 't' is 'TENNIS'.
Can you help me to learn more?
(Other definitions for tennis that I've seen before include "Sent in, oddly, to play a sport" , "Medieval, originally French, sport, codified by the British" , "Sport on a court" , "Game of the Williams sisters and of Roger Federer" , "Game played on court" .)