TL;DR:
- Russians have pushed several km past the northern border
- Not nearly enough Russian troops committed for a successful attack on Kharkiv, but that still seems to be their plan
- Lots of mechanized units coming across the border and getting drone-striked
In related news, on Monday it was announced that the commander for the region was replaced, without explanation, and allegedly some things have improved.
https://theguardian.com/world/article/2024/may/13/russia-tactical-success-advance-on-kharkiv-ukraine