Which is the correct one:
- On the Wednesdays of September, I used to go to the cinema with my dad.
- On September Wednesdays, I used to go to the cinema with my dad.
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On September Wednesdays
would be understood, but I think a native speaker would more likely use in:
Also, when referring to a set of recurring days of the week (Sundays, Mondays, etc.), one does not ordinarily use the definite article.
On Wednesdays in September, I used to go to the cinema with my dad.
Both OP's versions are completely unacceptable. In principle there's nothing inherently incorrect about the suggested alternative on Wednesdays in September, but it doesn't sound like very natural phrasing to me (I feel the way it emphasises plural Wednesdays is "odd" in this context). I suggest something like...
In September I used to go to the cinema with my dad on Wednesday.
It's grammatically optional whether you pluralise Wednesday there. I tend to prefer the singular because we're not actually talking about very many trips to the cinema in total, and we're already stretching the idiomatic "I used to" with such a small number of occurrences. Using the singular dayname seems to me to help "balance" the utterance. But that's really just a personal stylistic preference.