A tough question, because the literal translation does not work here at all. It is necessary to know the “context” to be aware of the real meaning, i.e. what is behind of this expression. My attempt of explanation: In Spain, Galician people are considered (please note: I am not judging at all, I am just describing the situation) to be “vague” in their decisions, opinions, replies to compromising questions, i.e. it is difficult to know what they really think, to get to the very core of their soul. So some Spaniards (not Galicians, obviously) say, if someone responds a question in a vague way, without really getting involved into the discussion, they are answering “a la gallega”, i.e. in a vague or elusive way, which means, without really getting to the point.

If you have something comparable in English, it is something I don´t know, but this is the explanation and meaning of “contestar/responder a la gallega”.

All this being said with the highest respect for Galician people and their region (where I spent a part of my adolescence).


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Note added at 33 mins (2021-10-08 16:07:15 GMT)
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Here just an example:

https://www.esdiario.com/medios/634923774/La-respuesta-a-la-...
Feijóo no se ve en Madrid pero responde 'a la gallega' sobre la foto de Colón

Alberto Núñez Feijóo ha evitado pronunciarse sobre si consideraba un error la fotografía de Colón, en la que el PP se manifestó junto a Ciudadanos y Vox contra el Gobierno de Pedro Sánchez, y ha tirado de la proverbial retranca gallega para contestar con un escueto "depende". Otra cosa ha sido cuando le han preguntado sobre su salto a la política nacional. Ahí sí que ha sido cristalino.


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Note added at 1 hr (2021-10-08 16:36:14 GMT)
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Well, it seems I have to reinforce my own response: "Responder a la gallega" may or may not imply asking another question. This is not at all the key point, but the elusive way of the reply, either by avoiding a compromising answer (see my example/quote) or by asking an elusive question. As simple as that. But to respond to a question with another question is NOT what it means, not always.