Martha Wainwright, Jazz Cafe, London

For once, the guest really is special. The unannounced treat at Martha Wainwright's first show since her son Arcangelo was born two months premature in November is his granddad, Loudon Wainwright III – aka the "Bloody Mother Fucking Asshole" of the savagely raging song which made Martha's name. "We won't be singing that song tonight," she assures him. Its fury at his parenting had clearly blown itself out anyway, even before grandparenting became the new priority, with Martha guesting on Loudon's latest record and at a New York show. Still, when he strides onstage, the healing hug you might expect becomes a professional passing of an acoustic guitar. They do embrace when he leaves, détente achieved.

"I'm just coming back into society," Wainwright says of this impromptu two-night stand. Arcangelo remains at the University College Hospital and she is still frazzled. "I'm totally spacing where I am," she confesses as she loses the thread of "Bleeding All Over You", one of a couple of stumbles in an otherwise full-force show. Even her garish tights were "inspired by the clowns at the children's ward".

Family business gives tonight a loose, intimate feel, as it catches Wainwright in the middle of huge personal change, but her music remains a story in itself. Its early-1970s English folk tunings and her voice's warm grain recall Nick Drake. Southern Californian singer-songwriters of that era are referenced too, in contrast to the pop sheen of her last studio album, I Know You're Married But I've Got Feelings Too.

The slippery, unpredictable chords of her melodies are all her own. Her voice's supple range and precise power, capable of thunder or a thin, tense croon, makes even the mysterious mix of religion and border drug-bust ambience of "Jesus & Mary" into a carnal drama. And when a pianist arrives for a selection from her recent album of Edith Piaf songs, their martial bar-room sway, so rhythmically and melodically different from Anglo-American folk, is an effortless switch. Piaf's persona fits Wainwright – she hunches forward, clutching her head, gesticulating and falling away from the mic.

And then there's Loudon. He's a big, imposing man with a ferocious grin, and Martha leaves the stage to him for a couple of songs. "This song I wrote before I had any babies," he notes of "Be Careful, There's a Baby in the House", from 1971, although they are its subject. "If your I love you is an IOU, don't expect to have a good deal," he sings. "Your pride and joy will not be had." It sounds like a warning to himself never to become the sort of dad to whom a daughter would one day sing "Bloody Mother Fucking Asshole". But the balance of power, and affection, is changed as Martha comes out to teach the old-timer to auto-tune his borrowed guitar. In the Wainwright dynasty, music is one currency of love. They duet as a playfully warring country couple: happy families, at last.