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Wednesday night’s contest amid bitterly cold temperatures at Prenton Park was settled by second-half finishes from Filippa Angeldahl and Mary Fowler.
The defeat means Matt Beard’s Reds must wait for other fixtures in the competition to be completed to see if their tally of nine points is enough to reach the quarter-finals.
City began to assert control of the ball, though without truly concerning the home team, with Hayley Raso’s driven cross-shot past the front of goal the closest they came during that spell.
The Reds therefore drop to second in the table, behind City, and with only the five group winners and one best runner-up to qualify must await results from elsewhere to confirm their fate.
Neither side was quite able to find a rhythm upon the restart, but City steadily began to exert a level of pressure that would yield the breakthrough before the hour mark.
The delivery was initially headed in at goal by Alanna Kennedy and blocked, but the ball was recycled to Angeldahl, who turned and dragged a low shot across Cumings into the bottom left corner.
Substitute Lauren Hemp sped dangerously through the left channel to create space for a low cross that was clinically dispatched first time when met by Fowler inside the box.