The north African side would have been bitterly disappointed with the result, having had numerous chances to put the game to bed before the Stallions' late smash and grab kept them in the tournament.
Moyes, in his first season in charge, has guided the Premier League champions to the last 16 of the Champions League as well as the League Cup semi-finals.
Barkley, 20, has excelled for the Toffees this season, but suffered the injury during his side's 4-0 FA Cup third-round win over QPR on 4 January.
The 32-year-old forward joined the Latics in the summer for around £2m from Norwich but only scored twice.
The Portugal captain, 28, was named as the world player of the year for the first time since 2008 by national coaches, captains and journalists.
The British number one was beaten 6-3 6-0 by Belgian 18th seed Kirsten Flipkens in 50 minutes in Melbourne.
Puncheon, who played under Warnock while on loan at QPR, tweeted the claims about the 65-year-old while responding to criticism of his penalty miss in the 2-0 defeat by Tottenham.
Edin Dzeko slid in to give City an early lead and control of the contest.
Carlo Ancelotti's side moved within three points of the top two, despite a disjointed display against Espanyol.
The 20-year-old England international sustained a suspected broken toe in last weekend's FA Cup win over Queens Park Rangers and sat out Saturday's 2-0 defeat of Norwich City in the Premier League.
The Premier League champions lost to Tottenham Hotspur on New Year's Day and that result was followed by an FA Cup exit at Swansea before more cup misery came in the shape of a 2-1 reverse in the first leg of their League Cup semifinal with Sunderland.
- Toure Praises John Obi Mikel
- Toure Gets Africa Player Of Year Award
- Pregnant Ennis-Hill Out Of Glasgow 2014
- Dauda Gets Afranie Backing
- Ibenegbu Set For Departure At Heartland
- We Are Playing Against Officials - Moyes
- Valencia Strike Late To Hold Atletico
- Morocco Aiming For Semifinals
- 'Unusual' Ghanaian Interest In Brazil
- Nigeria To SA Without Keshi
- African Experience Vital - Igesund
- David Moyes: Manchester United Boss Has Doubts On Transfer Targets
- Andy Murray To Go For Fourth Aegon Championships At Queen's
- Sam Allardyce: West Ham Owners Back Manager
- Laura Robson's Australian Open Preparations Suffer Setback
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