Private Information Retrieval and Writing at the same time possible?
Here's a single-server one: https://link.springer.com/content/pdf/10.1007%2F978-3-642-13708-2_26.pdf

Obviously the data would need to be encrypted, otherwise it will be obvious what was changed, and the PIR-write protocol needs to somehow re-randomize the ciphertexts that were not modified.