WILL he, won’t he, and when will he eventually get round to telling us? Since his reluctant exit from the White House, Donald Trump has managed to keep America and the world guessing about another run for the presidency in 2024.

Katty Kay, US special correspondent for the BBC, considers what it might mean if he does in a new documentary, Trump: The Comeback? (BBC2, Tuesday, 11.15pm).

As she shows, The Donald has never really been away, using his fundraising power and influence to help certain candidates, some of whom are hoping to win office in the midterm elections on November 8. What happens then will shape the second half of Joe Biden’s presidency and determine if he, and Trump, will restage their battle of 2020.

Kay begins her travels across America in Arizona, which used to be solid Republican until Biden pulled off a shock win. As Kay says, this started the ball rolling on Trump’s false claims that the election was “stolen” from him.

Her first stop is a shop selling Trump merchandise. Here, beside the usual MAGA (Make America Great Again) baseball caps there is a cardboard cutout of the former President as Rambo, yours for $100. The store owners are among the majority of Republican voters who believe the stolen election claims. If it happens again in November or 2024 there could be trouble, the couple fear.

It is a warning Kay hears several times as she drives from state to state, interviewing candidates, Democrat and Republican. Kari Lake, hoping to be the Republican governor of Arizona, used to be in the news business like Kay. The former Fox TV presenter gave up “immoral” journalism for politics. Lake is an election denier, and whatever counterargument Kay puts to her is batted away, no hesitation.

With emotions running high following the Supreme Court’s overturning of the law on abortion, a huge turnout is expected for the midterms, which used to be thought of as a non-event by many voters. Not this year.

Four years ago Vicky McClure set out to raise awareness of dementia, secure better services for those living with it, and generally sing out about the power of music in relieving some of the burden of the condition. I’d like to report that the Line of Duty star secured vastly improved services and never had to campaign again, but, well …you know the rest.

Our Dementia Choir Sings Again with Vicky McClure (BBC1, Monday, 10.40pm) takes up the story again. McClure begins by visiting some of the people who took part in the first series. Among them is Mick, of belting out “Buttercup” fame. Diagnosed with dementia four years ago at the age of 51, the change in Mick’s condition is marked, yet the family is still struggling to get the help they need.

McClure’s nana had dementia, and when the actor speaks about the condition, and the plight of carers, it is clear this fight is personal. At the same time, she wants as many people as possible – particularly those in power – to see what a difference singing and music can make. Last time the task was to put a choir together. Now the aim is to record a song -written specially for the choir, What’s Your Story, and get it to number one.

The end is nigh in every sense in Rise of the Nazis: the Downfall (BBC2, Monday, 9pm). After three years of outstanding documentaries mixing expert analysis with drama, the new run of films opens in December 1944. “Germany is losing the war on all fronts” a caption tells us. It is a calm introduction to the madness that follows.

As ever, it is the smallest details that linger longest in the mind, such as all the blinds being closed on the train bringing Hitler back to Berlin for the last time, lest he see the terrible destruction of the city at the hands of allied bombers. As the first of three episodes ends Hitler is holed up in his bunker as his henchmen busy themselves with thoughts of what will happen next. The first two series are still on iPlayer.

New sitcom alert, yes another one, for I Hate You (Channel 4, Thursday, 10pm/10.30pm). This comedy, however, comes with the distinction of hailing from the laptop of Robert Popper, writer of Friday Night Dinner and Peep Show.

As in Peep Show, I Hate You has at its heart the relationship between two housemates. This time, in place of warring Mark and Jez (David Mitchell and Robert Webb) it’s Charlie and Becca (played by Tanya Reynolds and Melissa Saint), a couple of twentysomethings who actually like each other.

The first few episodes don’t have the gag rate of Popper’s previous work, but Reynolds and Saint are a real find.