Brough, a popular figure in local football circles, died just three days after his appointment as club chairman and director at the age of 45.
In a statement, the club said: "Our thoughts and condolences go to Jamie’s wife Karen and his entire family at this incredibly difficult time.
"As a mark of respect to Jamie and his family, the club will be leaving the position of chairman vacant for the 2022-23 season."
A minute’s silence was held before two games played at Basford United’s Greenwich Avenue stadium on Saturday.
Club owner Chris Munroe tweeted: "Very poignant day packed with sorrow and thoughts for Jamie's family.
"Great win for our Basford United Academy to lift the cup. Busy day off the field as we try to resume normality.
"Some great new player meetings with Mark Clifford and finally a good feel to our Step 3 destiny."
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