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Match referee Evan Boyce took no action at the time but notified the IFA's Disciplinary Committee regarding the incident following the Solitude game.
It considered the correspondence together with video footage.
Oman was charged with a breach of Article 18.11 of the Disciplinary Code (assault or battery of an opponent).
Portadown were also fined £100 by the Disciplinary Committee.
Cliftonville won 1-0 in what was Niall Currie's first game in charge as Portadown manager.
Portadown defender Ken Oman has been suspended for six matches for elbowing Cliftonville's Caoimhin Bonner in Saturday's Premiership fixture.