Social science/humanities PhD in UK: Take partially funded PhD, or work for a year and reapply for full funding?
"self-funding" just means you're responsible for paying that part, whether it's fees or your day-to-day living costs. They don't care whether that's from part-time work or previously earned money, or from getting one of the other scholarships you're applying for.


I will add as a caveat they may ask for proof of your ability to self fund, but since you say you have savings / family support and the dept itself is offering teaching hours this shouldn't be an issue

PhD funding in the art/social sciences/humanities is difficult to obtain in the UK, getting full funding will be eye of the needle stuff from what I get told 
I've never seen a CV where the phd funding source was asked for, I don't mention it (EPSRC) on mine
Can't make that choice for you. I can say I gave up a job for a phd (less money) and my circumstances sound similar - i.e. young and no family to support.


One question I have is what is the situation with tuition fees? At one point you mention having a tuition fee waiver - is that a full waiver or part? It makes a difference, if you had a full phd scholarship in the uk you'd expect your tuition fees paid and a 15K annual stipend. If you're say 50% funded, you'll get a 7.5k stipend which is tight (depending on where your university is, I did mine 20 years ago and the full stipend was then only 5K :-( ). Anyways, if the tuition fee waiver is only partial, they will expect you to pay half of it, that would be typically around half of 4K (2k). If that comes out of your half stipend, you're down to 5.5k - or what I was struggling on 20 years ago. Your 400 a month for teaching might buoy you up a bit, but bear in mind this will be term-time only. So this is the economic question. If you can afford that, then it's down to where you really feel you want to be - getting a phd or working. Don't overlook the fact social life is usually better at university - a big part of what made me go back.

Also re: the job, if your part-funded phd would start in october, well you'd be handing your notice in as soon as you arrived if it's a typical 3 month notice period. I'll hazard a guess though that they have a probationary period for new starters (most places do) so they could wash their hands of you at next to no notice at any time in the next 6 months or so. And guess what, you can do the same. My attitude is, if that's the case you might as well take the 3 months earnings.