Obscure cricket side, Irish, impresses head office (3-5)
I believe the answer is:
one-horse
'obscure' is the definition.
(one-horse can mean obscure or small-scale)
'cricket side irish impresses head office' is the wordplay.
'cricket side' becomes 'on' (cricket term, on or leg side).
'irish' becomes 'erse' (I've seen this before).
'impresses' indicates putting letters inside (some letters pressed into others).
'head office' becomes 'HO'.
'erse' placed around 'ho' is 'ehorse'.
'on'+'ehorse'='ONE-HORSE'
(Other definitions for one-horse that I've seen before include "(Of a town) having few facilities" , "Small and remote, town perhaps" , "Small town, with only one quadruped?" , "(Of a place) poor and lacking amenities" , "With only a single equine, the town is very small" .)