[EDIT]
Sorry, didn't see your latest post before I wrote this
[/EDIT]
"I gave her 3k, which is the max you can give in a tax year without incurring tax."
It's rather more nuanced than that.
There is no "gift tax" in the UK - apart from inheritance tax (IHT) considerations if you die within seven years of making a gift.
So, for example, let's say that you gift somebody £5k in a particular year. You then die six years and 364 days later.
What happens in that situation?
Well, if your executors even notice that amount then it will mean that you are £2k above the limit for that year.
What is the practical consequence of this?
If the executors even notice this amount then it will mean that the tax free allowance you get for your estate of £325k per person or up to £500k if you own a property. Or double those figures if you are a surviving widow/widower and your deceased spouse left everything to you.
So, if you own a home by yourself, instead of being able to pass on £500k tax free you would only be able to pass on £498k tax free (or £998k tax free if you are a widow).
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Rather off-topic but this issue can be more of an issue for the giftee.
The whole thing about it being taxable is much more an issue for the person receiving the gift, especially if you are giving them money on a regular basis.
If you are just giving relatives money out of your regular monthly income to help support them then it is no problem at all. You can give them as much as you like each month provided that you can afford it without affecting your own lifestyle. It will be excluded for IHT purposes and they won't be treated as receiving an income either.
Other situations can all get very complicated and messy.
Let's imagine a wealthy woman who has a younger lover, a "gigolo" if you like.
She helps to support his lifestyle by paying off his credit cards or rent etc and, in return, expects nothing other than ...
In this situation, HMRC would count those as gifts that the gigolo is receiving. However, if the gigolo has several, similar, relationships at the same time (or one shortly after the other) then HMRC would much more likely look at that as income and so expect to receive their cut.