If you are in support like that, you generally do need to be at work exactly on time. Same as being a cashier, factory worker, etc.
It's not like being a programmer, sysadmin, etc where exact hours typically don't matter much as long as you're available for meetings and the work gets done.
I just don't buy into the whole 11 workers do 10% more work than 10, or that 8 hours is more productive than 7.5. In any business that is, not just "brain work". Far-east sweat shop exempt.
I was in good terms with everybody, managers as well, and I think in that stupid age I did some brilliant work I'm still very proud of. After a couple of years, when all the rumors about me being hired fulltime were materialising (I was working for them through an agency), they just decided to let me go.
It hurt at the time, but it was a good decision that served me as a lesson[1].
Employers, unless you're a bank with set opening times, don't fret over 10 or 20 minutes. Programmers are productive only in a stress-free environment. Obsessing over 10 minutes is the total opposite of that. But please fire engineers like me, you'll do them a favour.
1: I stopped being so late in future jobs, but I also discovered I have some kind of DSPS. It's still bad, but I'm now a freelance (in Europe) working for US companies, my clients are having their coffee when it's 3pm here.
I haven't been diagnosed, but I've struggled with falling asleep at normal hours my whole adult life: I've tried sleep deprivation, advancing my sleep cycle[1], pulling all nighters to feel really tired in the evening and go to bed at an acceptable time, but it works only for a couple days, then I'm feeling sleepy a hour or two later than the day before, until I'm back going to bed at dawn. It's like having a circadian rhythm of 27 hours, instead of the normal 24/25[2]
The only thing I haven't tried is melatonin, but apparently in the UK you need a prescription to buy it.
1: Basically going to bed later every day, until you go around the clock and are going to bed at the desired time. Not applicable if you have a full time job -- https://en.wikipedia.org/wiki/Chronotherapy_(sleep_phase)