It has not been possible to positively identify this man. He does not appear in the records of the Commonwealth War Graves Commission as a casualty, nor those of the Regiment.
The 1901 Census does, however, record a Henry Lovery, then aged 14, who had been born in Stockport and was living in the town, working as a cotton bobbiner.
The Census also notes John Lovery, aged 49, an engine house labourer and Mary, aged 41. These were probably his parents. Also recorded was Martin, 18 and Thomas, 6. He is almost certainly related to Ulic Lovery, also commemorated on the Memorial.
Assuming this is the same man, then he would have been aged about 27 when War was declared and 31 when it ended. He is certainly in the age group to have seen active service.
The National Archives on-line "medal entitlement" database lists a Henry Lovery, who served as a Driver with the Army Service Corps, so it is possible that the regimental description on the Memorial has been wrongly inscribed. Alternatively, if this is not the man, then Stockport's H Lovery may never have served overseas and, therefore, would have no medal entitlement.
There are a number of explanations why he might not appear on the War Graves Commission records. The most likely of these is that he died some considerable time after the war ended and the family simply did not notify his Regiment. Possibly, if the family thought he was with the Cheshires and not the ASC, the paperwork may have been lost. Alternatively, he may have died for a reason that the Commission felt was nothing to do with his war service (and, as such, would not include him in the records). With the passage of time, it is impossible to know what happened.