Wright's strong start to life at his new club continued as they beat Fleetwood Town 1-0 in their third round FA Cup replay on Tuesday.
The 24-year-old defender played the full 90 minutes as City moved onto the next round and a trip to Burnley, who also came through a replay.
Burnley, whose young Aussie midfielder Aiden O'Neill came on for the final few minutes, beat Sunderland 2-0 at Turf Moor thanks to goals from Sam Vokes and Andre Gray.
But there was late heartbreak for Ryan Williams' Barnsley, who conceded in the injury-time of extra-time to lose 2-1 against Blackpool.
Williams came on with 15 minutes left in normal time with scores tied at 1-1 and almost won it for his side in extra-time.
The 23-year-old hit a powerful shot from just outside the box that the Blackpool's goalkeeper managed to parry away to safety.
111 - CLOSE AGAIN! Ryan Williams has a pop from 20-yards, Slocombe pushes it away to Watkins, who sees the effort cleared off the line #COYR— Barnsley FC (@bfc_official) January 17, 2017
But just as it looked like the match was headed to penalties, the visitors found a winner through Bright Osayi-Samuel.
Meanwhile, Aziz Behich's Bursaspor endured a tough night in the Turkish Cup, slumping to a 4-1 defeat against Akhisar.
The Socceroos left-back watched from the bench for the 90 minutes as Ricardo Vaz Te scored all four goals for Akhisar in their emphatic win.