Gavins shot an excellent six-under-par 64 for a two-round aggregate of 135.
Wolverhampton’s Aaron Rai leads by two after carding a six-under-par 64 for his opening round and signing for a 68 in the second.
Gavins shot three birdies on each nine in his second round to head a group of five Yorkshire players who made it through to the last day’s play.
Hallowes’ Scott Campbell (71 70) is tied 27th after matching three bogeys with three birdies.
Alongside him is David Booth (Rotherham) who shot the same scores as Campbell but in the opposite order. He had 17 pars and a bogey in round two.
Also on 141 is Middlesbrough’s Michael Skelton who lowered his first-round 73 by five shots.
After going out in two over he had a storming finish, birdieing four of the last five holes.
GRENKE Championship winner Marcus Armitage (Howley Hall) made it through in a tie for 47th, going in the opposite direction to Skelton by sliding from a 68 to a 75.
He only had one birdie in his second round having grabbed five the day before.
He was on the cut line but among those below it were Chris Halley (Malton & Norton) 69 75, Paul Lockwood (Hessle) 72 73, Joe Heraty (Moor Allerton) 72 74, James Maw (Sheffield) 72 75, Steve Uzzell (Hornsea) 79 70, Graeme Clark (Doncaster) 73 76, Adam Walker (Huddersfield) 79 71, Richard Law (Wakefield) 73 77, Tom Perry (Abbeydale) 75 76 and Nick McCarthy (Moortown) 77 76.