@AllThatFancyPaintsAsFair
No need to be rude. As I said, I'm not sure exactly how it works for TAs starting mid-year. What I do know, however, as I work in education for a local authority, is that TAs are not salaried workers so your point it irrelevant. It is not an annual salary, it's an hourly (totally exploitative) wage with a casual contract made to look like an annual salary - she is paid for 39 weeks work and they have given an equivalent annual salary. Her paraphrased quote about the terms of employment says this quite clearly. It is also absolutely the standard employment situation for TAs. www.tes.com/jobs/careers-advice/teaching-assistant/teaching-assistant-pay-and-conditions.
Essentially, with term-time only contracts, TAs are unemployed for the holidays (but can't claim unemployment benefits, so pretty screwed).
OP will probably have a 6 month contract - her contract will be need to be renewed before September - she will work and be paid for March, two weeks of April, c.3-4 weeks in May, June and 3-4 weeks of July + 6 months of annual statutory holiday allowance (14 days). So c.20 weeks. If this is split across the 6 months, so she gets a pay packet every month instead of nothing in August (which is what happens in the USA for teachers and used to happen in U.K. too), then it's 20 weeks pay split across 6 months (20/26). The new contract from September will pay her the full amount originally quoted, split over 12 months. 39 weeks pay + 28 days statutory holiday spread over 52 weeks (43/52) which will work out slightly more. Not to mention the fact that actually the holiday entitlements are likely to be less than the full statutory amount as the contract is not for a full year (as I said, TAs are often screwed over). It's more likely that she'll get 21 days (39/52 * 28 days).
When I put this amount into a tax calculator based on an hourly wage of around 10.40 ph (calculated from her 26.25 hr week over 39 weeks and adding in paid holidays), I get just over £800, which is not far off what OP has been paid and suggests that this is quite likely to be what has happened. Or it could be tax codes. Or both.