Flimby were first to score as David Torley, later named man of the match, crossed after only 12 minutes and Calum Moffat landed the extras.
But Kells stormed back to lead with an unconverted try by Jordan Herve and a second touchdown from Kieran Alcrow, which he converted himself.
Flimby finished the half on top and two tries in as many minutes, just before the interval, gave them the edge.
Darren DeGraff and Torley both crossed but neither was converted.
Gradually Flimby pulled away in the second half after Moffat scored a try and converted it on 52 minutes.
Just after the hour Darren Pape went in and again Moffat converted, but the final try on 63 minutes by Bradley Kirkbride was unconverted.
Veteran Lee Rutherford picked-up the man of the match award for Kells A.
That turned out to be the only game played in the league, although two others were scheduled.
The Great Clifton field was unplayable for their clash with Hensingham A and after arranging to borrow the Broughton pitch were disappointed to have that also ruled unplayable.
The other game between Aspatria and Seaton A was called-off because the visitors could not raise a team.
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