Kent began on 70 without loss in their second innings, still 126 behind and they suffered a major blow went Tawanda Muyeye went to the fourth ball of the morning when he edged Abbott to Toby Albert at second slip and was out for 46.
Abbott then removed Ben Compton for 25, again caught by Albert, before Abbas took two wickets in nine balls, Daniel Bell-Drummond for 23 and Joe Denly (five).
Abbott captured Joey Evison for 11, victim of a low slip catch by James Vince, leaving Leaning and Finch to survive to 145-5 at lunch.
The duo batted out the entire afternoon session with Hampshire’s tactic of bowling short backfiring as both batters comfortably dealt with a barrage of bouncers.
Leaning glanced a single off Abbott to bring up his century but he was out soon after, ending a record sixth-wicket partnership for Kent against Hampshire and reigniting the contest.
Charlie Stobo (17), Matt Parkinson (seven) and George Garrett (four) followed but Jas Singh lingered for 27 balls in a crucial last wicket stand with Finch that used 11 overs and ended only when the latter tried to hook James Fuller and was caught at first slip by Vince.
In bad light and with nine fielders on the boundary the run chase was largely theoretical, especially when the rain returned.