I'm pretty sure that in UK retail banking, this used to matter but nowadays, as a general rule, does not usually matter any more.
if I open the account there will this account be registered with that city b branch?
Will this in anyway affect what can be done with the account, assuming that visiting the city b branch is usually very awkward for me?
Never in nearly thirty years of being a UK retail banking customer has it mattered which branch my account was held at. The days of one's own 'bank manager' being someone to have a personal relationship with are long gone - these days it's all algorithms and 'personal account care consultants', or whatever grim corporate-speak your particular bank uses.
As an extreme example, all customers of First Direct have the same branch address, since it is an online-only subsidiary of HSBC.
For (mutual) building society's banking operations things might be different, but I don't think so.
I will finish with a caveat: the one exception I know of. In 2013, when Lloyds Banking Group was forced to divest a chunk of its retail banking operations, a number of customers who had opened accounts with a branch of Lloyds Bank (before it became Lloyds TSB) found that, as part of the divestiture, they were going to become customers of the (newly-resurrected) TSB Bank, rather than remaining with the portion that reverted to being Lloyds Bank. The division was made on sort codes, which ultimately come down to the branch the account was opened at.
Not sure anyone cared though.