Initially, England and Australia support cricketers providing home for rescued dogs (9)
I believe the answer is:
battersea
'home for rescued dogs' is the definition.
I can't judge whether this definition defines the answer.
'initially england and australia support cricketers' is the wordplay.
'initially' says to take the initial letters.
'and' is a charade indicator (letters next to each other).
'australia' becomes 'A' (abbreviation for Australia).
'support' is a charade indicator (letters next to each other) (in a down clue, some letters hold others up).
'cricketers' becomes 'batters' (I am not sure about this - if you are sure you should believe this answer much more).
The first letter of 'england' is 'e'.
'e' next to 'a' is 'ea'.
'ea' after 'batters' is 'BATTERSEA'.
'providing' acts as a link.
Can you help me to learn more?
(Other definitions for battersea that I've seen before include "where dogs' home is located" , "Site of well-known dogs' home" , "where best friends might end up" , "London park" , "Area of SW London" .)