At the end of Chapter 4, we learn that "a relation" of Mrs Dashwood's "understood that she was in need of a dwelling," and offered to rent her a cottage on his property. Mrs Dashwood feels that "the whole of his letter was written in so friendly a style as could not fail of giving pleasure to his cousin; more especially at a moment when she was suffering under the cold and unfeeling behaviour of her nearer connections. [emphasis mine]"

Her nearer connections are John & Fanny (& little Harry, of course). These nearer relations are also closer (nearer) in proximity at this point in the story. But as soon as the Dashwood women move to Barton Cottage, her more distant relations become closer, location-wise as well as in attitude and general familial support.

I realize that "cousin" doesn't necessarily mean first cousin (or even cousin at all), but Sir John Middleton obviously feels that any relative is worthy of notice.