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So what are these "mysteries of the Faith which MUST necessarily be known and believed" that St. Pius X was referring to here?

Vatican I puts that nail in the coffin of Rewarder God theory by defining supernatural faith as something that has for its object a truth that can be known ONLY by Revelation.

So what are these "mysteries of the Faith which MUST necessarily be known and believed" that St. Pius X was referring to here?

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Given that the quote is actually from Benedict XIV, the answer would probably be found in his (rather than Pope St. Pius X's) corpus.  The footnote in the encyclical indicates a docuмent shorthanded as Instit. but I'm not sure what docuмent that is, and it could even be a private work of Pope Benedict's since he is largely regarded as one of the best theologian-popes and published many works aside from his pontificate.
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I know that in the context of Acerbo Nimis pope St. Pius X is considering primarily the poor state of education among Christians, and the paragraph where he cites Benedict XIV is one where he's just discussed how Christian faithful who had plenty of opportunities to learn, who were not weighed down by the obligations of their state, avoided all things religious and have tenuously left their religious instruction up to the priest who will be delivering them last rites.  So at least in that respect it is plain that the Pope isn't attempting to teach what articles of faith constitute the necessary elements for having supernatural faith, or even if any do.  It would be good, for these discussions, to see material which engages that point very directly rather than obliquely along the way to make a different point.