In Jane Austen’s letters is this, rather dismissive one about Byron:

I have read ” The Corsair, mended my petticoat, and have nothing else to do.”

Letter to Cassandra, sent from Henrietta Street: Saturday ,March 5, 1814.

We know,that as an avid reader of new literature/poems, she had read The Giaour and The Bride of Abydos by Byron, as she specifically mentions these, with other contemporary poems, in Chapter II of Persuasion, when Anne Elliot discusses poetry with Captain Benwick :

He was evidently a young man of considerable taste in reading, though principally in poetry; and besides the persuasion of having given him at least an evening’s indulgence in the discussion of subjects, which his usual companions had probably no concern in, she had the hope of being of real use to him in some suggestions as to the duty and benefit of struggling against affliction, which had naturally grown out of their conversation. For, though shy, he did not seem reserved: it had rather the appearance of feelings glad to burst their usual restraints; and having talked of poetry, the richness of the present age, and gone through a brief comparison of opinion as to the first-rate poets, trying to ascertain whether Marmion or The Lady of the Lake were to be preferred, and how ranked the Giaour and The Bride of Abydos, and moreover, how the Giaour was to be pronounced, he shewed himself so intimately acquainted with all the tenderest songs of the one poet, and all the impassioned descriptions of hopeless agony of the other;

As to JA and Byron socially:I’m sure they moved in very different circles.

With the exception of these two specific mentions, quoted above, I can’t find any other Byronic references specific to JA.

There is a JASNA Persuasions essay you might care to look at. It might throw some more light on research pointers.

Barry, Elizabeth. “Jane Austen and Lord Byron: Connections”, Persuasions: Journal of the Jane Austen Society of North America. 1986 Dec., 8, 39-41