I'm going to get flamed for this, but, why are you dancing to her tune ?
Her world is Tesco and playdates. Why ? Because that's what suits her, that's why.
Would you tolerate this from anyone else ? Basically dictating what will and won't be happening ? Seems you're suffering a bit from the dreaded, 'ought to'.
She doesn't work ? Why ? None of my business if you like, but, you've got no common ground except for what she wants, have you ?
She's from China, ok. Again, none of my business, but why did she want to settle in Britain ? Seems to me she's got no intention of integrating and quite a list of things she doesn't want to (chooses not to) do.
What does she do when not at Tesco or asking for a play date ? I'm tempted to ask how good her vocabulary is. You're not her cultural advisor.
Is she just parked in front of the t.v. at all other non-Tesco times ? What does she do whilst on these play dates ? Is it just the children that come ?
Again, I'm probably going to get flamed but it sounds a bit like she's using you as part of the cultural immersion (for her child at least) into British life whilst not giving anything back. How fluent is her child's English ? Could the child speak English before they arrived here ?
Do they speak their mother tongue at home ? I'm guessing they do. Sorry, but she sounds selfish to me.
Also, could you not just ask her to ask someone else and simply say, 'my kids are older than yours, they don't have much in common.' Or words to that effect.
Are these play dates always requested by her ? Basically, how much of it is child led by her kid ? Does the child not ask to see or play with any other friend or child ? Sounds like the child is overwhelmed by it all being so foreign. As well as having a mother who obviously wants very little to do with the outside world, except what benefits her.
Also, these play dates, must they be hosted by you ? Do you ever get invited by her ? Does she ever host either at home or elsewhere ? She wants her child to play with yours to improve their English and British social skills and it seems to me the child is really struggling.
Does, 'can we have a play date ?' sound more like, 'can I offload my child onto you for a while you feed and/or entertain them ?'
Put your foot down, don't be taken for a mug.