Hi, you could try removing the greasy patches by washing with sugar soap first, then repaint it. I personally use oil based undercoat and gloss mixed together 50/50 to seal things like this if I can't get rid of it by cleaning. With the undercoat it with it, it will give a duller finish rather than gloss or sheen which I find is easier to emulsion over afterwards.
By the way have you put this question on another site as I have just seen an almost identical one on a different site.
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