Boston Town seal place in next round of FA Vase
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Skipper Lee Beeson had out them 1-0 up with a pinpoint 1st half free-kick before Adams doubled their lead soon after break.
The Midland League side reduced the deficit before Adams added another to make it 3-1.
Ingles again pulled one back late but the Poachers held on for a deserved win.
They're now at home to Stone Old Alleynians in the 2nd round proper on Saturday 11 November.
Elsewhere at the club, hat-trick hero Charlie Rigby guided Boston Town under 18s to a superb 4-1 away cup win at Northgate Olympic.His goals saw the young Poachers through to the next round of the Steve Asher Memorial Cup.
Kacper Szymczak scored the other goal, Northgate had beaten Boston in the Lincs Intermediate League South 3-0 the previous week and were league leaders.