Heavy rain overnight and morning drizzle meant a major mopping-up job and play eventually started at 12.55 BST.
Leach’s second ball of the day saw Felix Organ, on 12, edge to Kasey Aldridge at second slip.
Albert, who had been dropped before scoring by Aldridge on day one, moved to a valuable half-century, while Aldridge was at fault again when spilling a straightforward chance at square-leg, offered by Ben Brown on 10.
Albert hit straight sixes off Leach and Shoaib Bashir as the seventh-wicket partnership took Hampshire into the lead before the opener was caught behind flashing at a delivery from Aldridge.
Brown departed for 35, caught at backward point aiming to reverse-sweep Leach, to leave them 159-8, but Somerset’s hopes of restricting the lead suffered a blow when Bashir dropped Abbott at mid-on off Leach.
Abbott scored 18 before driving a catch to short cover off Lewis Gregory.
James Fuller then hit Archie Vaughan for six over long-on before perishing in the same over, caught at deep extra-cover for a rapid 25.
Tom Lammonby managed only a single before edging a catch behind off Mohammad Abbas at the start of Somerset’s second innings.
Abbott pinned Vaughan lbw for 25 and then induced edges from Andy Umeed and Tom Abell to have the hosts 38-4.
The scores were level when further rain curtailed the day’s play.
Report supplied by ECB Reporters’ Network.