The OP asks what do you expect if someone says "I will deal with it".
The general synopsis in most dictionaries is :-
'To take ' action ' in order to achieve something or in order to solve a problem'.
If our passports were in need of renewing and my husband said " I will deal with it" I expect him to renew them. If a teacher was telling parents of a bullied child " I will deal with it " I would expect that teacher to take action and do as they said they would. I expect that action will be taken by the person who said it because they are making a commitment, a promise, a pledge and if they did not 'deal with it" I would be more than a little annoyed.
In the particular case of Corbyn and his statement over the 'historic tuition fees' debacle using his words "I will deal with it" he was at the time given a lot of media coverage and at that time both left and right media outlets viewed his words as going to take action / a vow, that is clearly evident in the headlines at that time. The evidence for that is clearly available. I use the words 'at that time' because obviously there has been a change in peoples mindset and media coverage.
At no time prior to the General Election did the question be asked of Corbyn if his words could not be taken seriously , the mindset was Corbyn is a man of honour / Corbyn will do as he says 'Deal with it'.
Fast forward and John McDonnell declared it was never a promise, even Labour MP's were doing the rounds saying it was going to be dealt with as their leader had said, Mc Donnell said it was only an 'ASPIRATION'.
It is true to say it was NOT in the Labour Manifesto, Corbyn did however make a strong headwind prior to the general Election with favourable coverage which Labour capitalised on , especially with the younger voters. Is he a liar, no. Can Corbyn be accused of a U Turn/ Back Pedaling , yes to some no to others.
Of course opinions will vary but to me it all boils down to it depending on whether or not you/we/I as individuals would say 'I will deal with it" and see that as a personal 'principal ' to abide by or say it believing it has no influence on our actions to the person we said it to.
I think " I will deal with it " when said by a politician is about as vacuous a statement as " Lessons will be learnt ".