The school are far more likely to be sympathetic if you are honest with them and consult them.
Remember that 'unauthorised' does not mean 'disapproved of', just means 'doesn't meet the guidelines for authorisation'.
Speak to the school and explain your dilemma. They will, quite reasonably, want to know:
- Why you have to go there, rather than your relative staying with you
- The role of the children's father (as a PP has said, if he 'has to be at work', the school will take a dim view, whereas if you have been separated for years and has not been involved in his children's lives, that will be different)
- Whether the absence could be prolonged, or repeated
Even if the school does not authorise the absence, they may well be sympathetic if it is obvious that no-one else can do the caring, no-one else can have your children and this is a genuine 5 day one-off.
However, it would be sensible to start thinking long-term about your relative's care - what will happen after you leave?
(As a teacher, I was allowed a couple of days' leave in term time to take care of an elderly relative after a fall, because it was clear that I was going to use that time to set up something more robust and lasting than me travelling 250 miles to do emergency caring. I would not have been allowed indefinite or longer leave simply to provide short-term care without any longer term plans)