Hughes' header came in a frantic three minute period that saw the scoreline go from 1-0 to 3-1.
The opening half hour was relatively quiet, with neither 'keeper tested by the opposing attack.
The deadlock was broken when a measured cross from Hughes was guided into the back of the net by Tate.
Ashley Harper called Grant Young into action on 37 minutes, but Young comfortably tipped his effort over the bar.
The second half started with Jack Divall being replaced by Patryk Miszkiewxz after suffering an ankle injury, before Matt Maclean extended United's lead on 56 minutes as he connected with Hughes' cross, but two minutes later Wick got themselves back into the game as Andy Chick got on the end of Harpers pullback.
One minute later, Hughes found space to head home a Jason Taylor header as he made it 3-1.
Wick pushed forward and looked to get back into the fixture for a second time and Harper was instrumental in setting up Pedrick to slot past Young and ensure a nervy finale, but United stood firm to hold on for victory