Borough’s reward for a 3-1 home win over Farnborough at the weekend - thanks to goals from Marley Ridge, Joe Lea and Bristol Rovers loanee Harry Warwick - is a visit from Southern League Division 1 South side Melksham on Saturday, November 9.
Melksham, who play a level lower than Gosport in the non league pyramid, defeated divisional rivals Basingstoke 1-0 last weekend.
In their most recent league game, Melksham led Moneyfields 4-1 at half-time - only to ship five goals in 21 extraordinary minutes and lose 6-4.
The Wiltshire club are currently fourth bottom of their league, with just two wins from 10 games, and they also have the worst defensive record with 28 goals shipped.
Gosport - unbeaten in their last seven games - banked £2,450 as a result of winning their first qualifying round tie. Another £3,000 will be collected if the 2013/14 Trophy finalists beat Melksham.
Bognor Regis have been drawn at home to East Thurrock United, after Pompey loanee Bradley Lethbridge played a starring role in Saturday’s 3-1 home win over Uxbridge.
National South and North clubs - including Hawks, the team Gosport beat in the 2013/14 Trophy semi-final - enter the FA Trophy at the third qualifying round stage on November 23, with National League clubs entering at the first round proper stage on December 14.