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1.
Many people have said that this word means 'leave it out' or 'don't bother'. We it does mean this but it is also used as a sign of agreement to a situation.
It largely depends on the tone
So it can either mean 'fuck that', 'can't be arsed' or on the other had 'yeah mate', 'lets have some of that'
If said in the negative term it would be said with a lowering in the pitch of voice towards the end of the phrase, if however said in the positive it would be said with a rise in the pitch of the voice towards the end of the phrase.
As well as this in the positive it is accompanied by the whip thing chavs do on the 'that'
Jim: Mate have u got dat new playstation game where ya like kill people
Ben: Yeah mate allow that (pulls off a whip)

OR

Benson: You know we gotta be up at 6am tomorrow!!!
Hedges: Allow that
by The Chicken Shack Crew December 17, 2005
 
2.
“Allow that” is a term used in agreement with someone or something. It can mean something like… “yeah mate, that’s what I’m talking about” or simply “I agree with that”. Some may confuse this term with “Allow it”, which means “let it be” or “leave it out mate”.
Nicko: listen to this mate, it's well good...
John: allow that bruv, thats a tune!
by 2+2=5 March 21, 2006