You don't service the deficit. Servicing the debt is part of government expenses. The difference between government expenses and income is the deficit. The deficit is more important than the cost of servicing the debt because it is the shortfall for all government expenses. As long as it is falling in relation to GDP (and, by extension, tax receipts), we can go on with our current levels of expenditure because it is manageable.
If I was my daughter I'd accuse you of 'mansplaining' Jeremy.
I fully understand the difference between deficit and debt, but they are related - typically a deficit can only 'serviced' by increasing borrowing - if you deal with it by increasing tax (or receiving greater tax receipt) or reducing spending then you are changing the level of deficit rather than actually servicing it.
But I disagree that deficit is more important than debt, and certainly not more important than the cost of servicing that debt. There are times in which it make absolute sense to run a deficit budget (indeed that is pretty well what every government has done every year for decades), and to borrow year-after-year to fill that income/expenditure shortfall. The key is whether the cost of servicing that debt (i.e. debt interest) can be managed and that is a combination of two factors in the short term - the amount of debt and the effective 'interest rate' of that debt. In the long run you can also factor in growth as a debt of £1billion (and the cost of servicing that debt) in 2024 would be a tiny proposition of GDP and of overall public expenditure, but would have been very significant decades ago. And in this respect a bit of inflation is actually your friend.