This is an odd situation, thanks to きく's unusual polysemy. I'm inclined to read it as 'ask', for two reasons.
First is simply Japanese word order. English is fine putting quotation verbs on whichever side of the quoted material, but in Japanese, since in most sentences the verb goes after everything else, it's much more natural to put it after the quoted material. You can do it the other way, sure, and that happens all the time, but it's mildly at odds with Japanese's general tendencies - you have to put it in a separate sentence, and it comes off sounding a bit more like 'He spoke. "This is his quote."' than 'He said "this is a quote"'.
Second is the semantics of the other two verbs in the sentence. Each of them describes an action that まる子 has some degree of agency in - 'peek through' and 'face towards'. 'Hear' is a bit out of place, in that it would be much less up to まる子 whether or not she heard - that's largely out of her control, while the facing and the peeking are quite voluntary. With 'hear', you get a sentence like 'Maruko peeked into the kitchen, faced her mum's back and heard (it)', which, while intelligible, is a bit odd. If the goal was to say that she actively listened, the author would almost certainly have chosen a phrase to highlight the intentionality of the listening (perhaps 耳を澄ました), as 聞く corresponds to English 'listen' only in the sense of 'listen to music/the radio/some other intentionally-chosen background noise'. If the goal was to say that she 'heard the following information', I might have expected something like ...向かって、これを聞いた.
So I'm fairly confident that it means 'asked', since it's quite natural for 'asked' but slightly unusual for 'heard' or 'listened'. The reason that it's not in the first clause is because of the sequencing of events - the peeking, facing, and asking happen in that order. The author probably gave the content of the quote ahead of time to avoid breaking the sequence into multiple sentences (implying a longer pause between the peeking/facing and the asking than was meant), and to avoid having the quoting verb come before the quoted material.