"Ay, a very bad business, indeed. A new sort of way this, for a young fellow to be making love, by breaking his mistress's head, is not it, Miss Elliot? This is breaking a head and giving a plaister, truly!" (Ch.13)

The meaning of Admiral Croft's expression "breaking a head and giving a plaister" seems fairly clear to me. It means doing someone an injury and providing them with the cure. But it's an odd expression, so I decided to do a little digging. It turns out to be a proverb in use at least as early as 1430 (see Wordsworth Dictionary of Proverbs, linked below). Not much the wiser, I had a look at some contexts in which it is used:

1609. Ben Johnson's play (a comedy) "The Case is Altered", Act3 scene 2.
In a group of friends, Martino and Onion are practice-sparring with cudgels when Martino accidentally bashes Onion on the head. Onion claims it is nothing serious. He asks one of the others to bring him a 'platan' (presumably a plantain leaf, which 'stays the too free bleeding of wounds' according to Culpepper's 'Complete Herbal' ), and demands another bout, against the advice of his friends. Martino offers him him a cobweb (used in folk medicine as a dressing for wounds to speed clotting).

Onion: "How! a cobweb, Martino! I will
have another bout with you. 'Swounds, do
you first break my head, and then give me
a plaster in scorn? Come, to it, I will have
a bout."

1631. "Celestina", a play by James Mabbe:
Pleberio, lamenting the death of his daughter, is railing against the cruelty of Fortune:

"Thou dost put out our eyes, and then to make us amends thou anointest the place with oil; thou breakest our head and givest us a plaster; after
thou hast done us a great deal of harm, thou givest us a poor
cold comfort."

1738. "Polite Conversation", by Jonathan Swift:
Swift satirizes the conventions and pitfalls of so-called elegant and witty conversation through three dialogues conducted almost entirely in aphorisms, puns, proverbs and hackneyed expressions.

The scene is a group conversing in a drawing room. Aiming at wit, Mr Neverout makes a remark which seems to criticize Miss Notable's appearance. She turns it off with a light remark Neverout apologises, but she is clearly miffed. A few lines later, trying to make amends, Neverout offers her a pinch of snuff. Miss Notable exclaims:
"What! you break my head and give me a plaister; ..." and the conversation continues.

You can read the on-line text of the 1783 edition of 'Polite Conversations' at Googlebooks. The relevant passage is on p.68-69. At first I could not make head or tail of it, but once I got my eye (and ear) in I thought it both amusing and perceptive. I think JA would have loved it.

The quaint spelling here of 'plaister' for 'plaster' makes me wonder whether Swift's book might have been JA's source for this expression. Does anyone know whether Swift's 'Polite Conversation' was one of the volumes in the Austens' library?

It may be that JA needed no particular source if the proverb was common currency in her day. It was certainly still around much later in the 19th century, when the popular English Baptist preacher Charles H Spurgeon included it in 'The Salt Cellars - Proverbs and Quaint Sayings', (1880 or perhaps earlier). He wrote:-

"You break my head, and then bring me a plaster:-

All very fine, but it would have been better if the wound had never been made. At the same time, if we have injured another accidentally, he ought not to resent our kindly endeavours to make amends. All we can say is:- "Take all in good part, whoever thou be, And wish me no worse than I wish unto thee."