I'm missing that one. I've got a 4-0 win against Spalding on 15th October followed by a 5-1 win against King's Lynn on 20th October that season in the East Anglian Cup. So Boston were due another game as they hadn't been knocked out, but I haven't got it.
Looks like the "Up the Posh" database is missing it as well! Apparently in the East Anglian Cup Peterborough beat Romford 2-0 on 26th April before going down 3-0 to Cambridge City on 2nd May.
Boston played Peterborough twice that season in the Midland Floodlit Cup - losing both matches. They went down 2-4 at home in front of 4157. Not a bad gate for a midweek match in an obscure cup competition!
Also amusing to note the Saturday Midland League games that Peterborough played either side of that East Anglian Cup game on 11th April. They played at Gainsborough on 9th April in front of 700 and the return game against Trinity on 16th April in front of 6548. Gainsborough have never had any fans!