Hohoho, do pardon me. Please don't worry about it!

Literal

Hohoho, excuse-me-[very polite-あそばせ]. Please-don't-mind-[polite-なさらないで下さいな]!

ごめんあそばせ is an extremely polite, old-fashioned feminine apology — あそばす (遊ばす) is an honorific auxiliary, making this a very refined 'excuse me.' 気になさらないで is the honorific negative of 気にする 'to worry.' 下さいな adds a gentle, ladylike request. The whole register evokes upper-class feminine speech or period drama.