Hugh Laurie (“House, M.D.”; “Veep”) is just about the only reason to get into “Roadkill,” writer David Hare’s quick-moving but often ridiculously convoluted drama, premiering Sunday under PBS’s “Masterpiece” banner. As a kind of post-Brexit tale of upper-level government machinations, “Roadkill” couldn’t be more cynical about the political process. In another time, that might have seemed noteworthy or even wickedly entertaining. Now it just sort of feels like a dull and expected subtext to, well, everything in our lives.

Nevertheless, “Roadkill” is British, and it’s something a tad different (and more randy) than the usual “Masterpiece” period-piece fare. The distance of an ocean, combined with the safety of fiction, might qualify the series as a sort of escape from all the other worries weighing on a viewer’s mind. Those with a high tolerance for antiheroic political thrillers will have no problem watching it — or figuring out its overall plot well before it’s revealed.

Laurie plays Peter Laurence, once a furniture salesman, now the transportation minister serving in the cabinet of a shrewd, Tory prime minister (Helen McCrory). Laurence is well liked by the conservative party’s populist wing, stoking his reputation and spilling his deepest-held beliefs via regular appearances on a radio talk show hosted by Mick the Mouth (Tony Pitts).

Laurence has just successfully won a libel lawsuit against one of London’s newspapers, after a reporter, Charmian Pepper (Sarah Greene), couldn’t verify the source of her story about his alleged attempts to privatize the country’s health-care system. (The disgraced reporter, who is struggling with alcoholism, loses her job. Acting on a hot tip, she takes off for Washington to prove that Laurence was conspiring with deep-pocketed American conservatives to do just what her original story had alleged.)

Away from Mick’s mic and other enthusiastic supporters, we begin to see that Laurence is driven mainly by raw, self-centered ambition, believing that he can spin his way out of any crisis. He’s also a philandering husband and inattentive father to his two adult daughters. It’s not all that remarkable that Laurie is able to play such a brazenly bad character with empathetic and even compelling results — it’s very much in Laurie’s skill set. You almost begin to believe in the purity of Laurence’s intent, until the scheming becomes too much to take.

The prime minister reshuffles her cabinet and promotes Laurence to minister of justice, just as Laurence has discovered that he is possibly the father of an inmate doing time in one of the prisons he now oversees. Subplots begin to pile atop one another as Laurence dodges one potential scandal and gets out ahead of another, using it to a publicity advantage.

The title refers to an unfortunate encounter Laurence has behind the wheel of a car; the metaphor is caught in the headlights, as characters begin betraying one another with an almost comical undoing of all the knots Hare has tied around them. The series never makes a clear case that it’s worth all the effort — or worth adding another deceitful politician to your life at this particular moment.

Strange to think that by the time “Roadkill” reaches its (somewhat cliff-hung) conclusion four weeks from now, its American audience will be — well, who knows what we’ll be up to? Watching television, I suppose.

“Roadkill” (one hour) premieres Sunday at 9 p.m. on “Masterpiece” on PBS. Continues through Nov. 22.

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