You’d need to submit a flexible working request and set out why you’d like to be considered a full-time homeworker. Prior to April of this year, the expectation was that the applicant (you) would have to set out how such an arrangement would be beneficial for the employer too but the law was changed on that and you no longer have to do this,
I would, however, make it clear that you’re prepared to attend the office for essential F2F engagements.
There are no special parental rights. Being a parent (even a single one - and I do have empathy for the difficulties that being a lone parent can pose as I am one myself) isn’t a protected characteristic. But you can explain the particular limitations that your circumstances present and they can then consider your case on its individual merits. If they decline it, they’ll need to explain why it’s not viable - and, if they’re half decent, may even suggested an alternative that may be more acceptable to you, even if not exactly what you asked for.
Ultimately, however, they’re not obliged to agree. They’d have a very valid point if they were to say that you shouldn’t have applied for a job so far from your home if you knew that office-working was an expectation you’d struggle to meet. And we’ve moved far enough on now from the pandemic to understand that the move back to attending the workplace is become more commonplace again.
My advice is to be upfront and ask for what you want. Be pragmatic; prepare for the worst and start looking for alternatives closer to home, but hopefully you’ll be pleasantly surprised.
All the best x