Mr. Cat and Eugly do not interact very often on the show.
What has been seen of their relationship seems to not be very good, as Eugly beats Mr. Cat up in episodes like "Let's Play Lady Luck" and "Let's Play Wrestling", sonce he tried to hurt Quack Quack. In "Let's Play Tutus Galore", Mr. Cat joins all the other characters in making fun of Eugly, and in fact it is mostly his and Stumpy's fault that she started to doubt herself. Also, in "Let's Play Party", he frames her for cheating on Quack Quack (as part of an elaborate scheme to get Kaeloo to kiss him).
However, Eugly seems to be genuinely forgiving, as "Let's Play Castaways" has her attempt to rescue Mr. Cat and Quack Quack from an island