In reply, Surrey reached 64 for one at stumps on day two.
Opting to bat first, Leicestershire's stand-in captain Darren Robinson (31) and John Muanders (27) gave the side a sedate start before Clarke earned the breakthrough by castling Robinson to end the 63-run opening stand.
Clarke pouched a return catch to dismiss Mongia and had two tailenders caught behind later in the innings.
Mongia faced 41 balls to score 20 which included a boundary.
Hylton Ackerman top-scored with an aggressive 58-run knock from 62 balls that included 12 boundaries.
In another Division Two match, Zaheer Khan's Worcestershire reached 444 for three against Somerset at Taunton.
Graeme Hick (151 not out) hit his 99th century for Worcestershire and with Ben Smith (121 not out), he put on 274 - seven short of the county record for the fourth wicket.
This was Hick's 129th ton in first-class cricket and first of the season.
So far, eight players have scored 100 first-class tons for a county and the last one was Geoff Boycott for Yorkshire.