Sir Howard Hodgkin bobs towards me like a cork floating on a choppy sea. He is limping, hobbling. While waiting for him to emerge from his afternoon rest, Andy Barker, his assistant, showed me a collection of Hodgkin’s latest paintings, the subject of his first show of new work in London since 1999.

The artist’s studio, a huge, white converted dairy with amazing light, is set behind his Victorian townhouse on a road near the British Museum. There is a central bank of scrap and materials, a table of brushes, a basement with shelves of books and two dusty armchairs. Barker conducts a massive game of show and not-much-tell, sliding canvas covers that conceal the paintings this way and that, and deflects the most innocuous inquiry – “How long does he work for in a typical day?” – with a “He’ll talk about that”. I wouldn’t be so sure. The 75-year-old Hodgkin, one of Britain’s most pre-eminent artists, doesn’t describe his pictures; they are “representations of emotional moments”, daubs and swooshes of bright, dramatic colour with titles such as Privacy and Self-Expression in the Bedroom and Degas’s Russian Dancers. Their shapes and details rarely directly allude to their titles.

Many interviews with Hodgkin mention that he is reticent and intimidating, and he is widely reputed to be irascible and a bit of a pressure cooker. Sir Nicholas Serota, director of Tate Galleries, says: “He is highly emotional. It is not difficult to make Howard cry, but it is not an affectation. His emotions are very close to the surface and are there in the depth of colour he uses in his paintings.” Certain subjects seem off limits – no one writes about Hodgkin’s private life, beyond recycling that he was a married father of two when he came out as gay in the late Seventies. Last year, the critics gave him a kicking for a career-spanning retrospective at the Tate, so it’s surprising that he’s entering the fray again with this show. But he’s “exacting”, as he puts it, and determined to gain recognition, which amazingly he doesn’t feel he has, despite his public, if not critical, popularity and the large sums paid for his works.

Some of the pictures take years to complete, like Ozone, which he started in 2004 and finished only last year. Serota tells me Hodgkin can sit for “hours, days, weeks” planning each painting. Four new works (Hodgkin has still to complete the fourth) are taken from the chorus to Home on the Range, a cowboy song he heard when he was eight. I ask Barker (tufty-haired, utterly discreet) if he enjoys working with Hodgkin. “Very much. I can’t unravel the mysteries of him. Twelve years is a long time, so I must enjoy it.”

Hodgkin is dressed in black with close-cropped white hair and has a rich, honeyed, lugubrious voice. He won’t say if he’s happy with the new works (“That would make me a hostage to fortune”), but admits that he has been working hard. Why?

“Old age,” he shoots back. “I think the time comes when you think, ‘Well there’s not much time left.’ When I was your age I thought time was endless and suddenly it becomes clear that it’s not.” We’re getting deep very quickly. Does he consider his own mortality? “Yes. I don’t have some terrible medical reason or anything like that. It’s just that there are so many things I want to have done and I haven’t done them all.” Julian Barnes, his good friend, has sent him a copy of his new book, Nothing to be Frightened Of, which is about death. It sums up Hodgkin’s attitude, too.

He wants to do “better work”, and “more expressive” work specifically. This is said so laceratingly that I ask if he is hard on himself. “I think all artists are. That’s not unusual. It sounds very egotistical, but I think I mind what I think more than perhaps what anyone else does.” As he ages, he dislikes “losing control, which of course happens. I haven’t so far lost any physical control of what I use when I’m working, but I can’t walk as far as I once could.” And the hobble? “I’ve been ill,” he says. “Do I look frail? Two years ago I lost my balance, which was a very unnerving thing. I was in a house in France and the floor was concrete. I was leaning on a marble table. I slipped and hit my head on the floor. My balance has mostly come back but not completely.”

Even when he’s having his daily afternoon rest he still paints and repaints, “in my head”. But he is happiest in his studio. “A picture is finished when it is finished,” he says when I ask about the duration of some of his works. “In the end the paintings subsume the subject.” Will he take me through any of the events that have inspired these works? “That’s something I always avoid doing, because it doesn’t tell you anything.” The picture does the job? “Exactly.”

Hodgkin always wanted to be an artist. When he was eight and staying with relations on Long Island during the Second World War, he visited New York’s Museum of Modern Art. “I was very happy. America was another world and I was a young boy.” He saw paintings by Picasso, Barnett Newman, Pollock and Matisse. The experience changed him. When he came back to England in 1943, “nothing was the same”: Britain’s wartime drabness was in stark contrast to the atmosphere of New York.

He was closer to his father, an art lover and plant collector, than his mother. “She wanted me to be a diplomat and I’m the most undiplomatic person you could want. I would have started wars and been totally useless.” There was no art in his family’s West London home; his inspiration came from teachers. On Long Island, his art teacher would hold up images from books and reproductions. At Eton he was taught by the “extraordinary” Wilfred Blunt, brother of Anthony, the Soviet spy. “He could have taught a programme of concentrated wickedness and we would have gone along with it.”

Hodgkin wasn’t an outgoing boy. He clenches his fists to his eyes. “I was withdrawn, but happy, muted, rather than outgoing.” His family history, which he was unaware of growing up, is blue-chip: his great, great-grandfather’s brother discovered Hodgkin’s disease; two of his cousins were Margery and Roger Fry, the celebrated prison reformer and art critic. Another cousin, Dorothy Hodgkin, won the Nobel Prize for Chemistry in 1964. Hodgkin was particularly close to Margery and her house was a riot of arty decoration and calligraphy on doors. “I wanted to become an artist, but it seemed impossible. My parents said, ‘You’ll never be able to earn your living. It will be ghastly. It’s much better to be an amateur.’ For me, unless you earned your living as an artist, you weren’t a real one.”

Hodgkin ran away from nearly every school he was sent to. “I remember being taken back by the police. One officer was very fatherly and said, ‘Why did you do this? Are they maltreating you?’ I said, ‘No, I ran away to be an artist.’ ‘Good for you’, he said. I hung on to that for some time.” His voice croaks as if he is about to cry. “I thought something would happen and it eventually did – I went to art school, which in my world was unheard of.”

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