Margaret
What is clear is that there is no surviving service record for your man (60% plus of them were lost in a WW2 bombing raid). While the medal cards can't be relied on 100% (they only record men who served overseas service), looking through them we find that all other cap badges can be eliminated, he is the only Serjeant (old spelling) Harry Trickett , his ASC number has a T/ prefix which means horse transport and his dress is that of a man who works with horses. The medal rolls for Trickett with initial H support that also.
As m'learned friends might say, on the balance of possibilities, your man was T/309046(ASC number) then GS/95504 10th Battalion Royal Fusiliers.
The roll for the 10th Battalion RF shows a number of sergeants from the ASC transferring in with consecutive numbers suggesting it was at the same time. There may be something in that clue worth looking at, I'll ferret around (or perhaps someone will actually know!).
MaxD