The visitors finished the day on 152-1 with a commanding lead of 216, giving them a platform from which to attack the Hampshire bowling and try to force an unlikely win during Thursday's final day.
Resuming on 162-3, Hampshire's middle and lower order all chipped in with useful contributions but no batsman could go on to contribute the big score needed to get close to Yorkshire's first innings total of 415.
For Yorkshire, Adil Rashid was the pick of the bowlers, taking 4-62 from just under 19 overs.
Hampshire limped towards 350 and an extra batting point and as soon as that target was reached – and with his side still 64 runs behind – Pothas declared the innings closed. Hampshire desperately need a win to climb off the bottom of the table.
Adam Lyth added a half-century to his century in the first innings, moving past 800 Championship runs for the season in the process, while Anthony McGrath showed the sort of form which saw him gain England honours in 2003 to smash a 66-ball 50.
Full report and scorecard in Thursday's Yorkshire Post.