I have heard people talk about DFT functionals built starting from MP2 corrections. Do you happen to have a reference I read about that from?
These are called double hybrid functionals. You add a fraction of MP2 correlation on top of the DFT energy.
A good reference is e.g. the review by Goerigk and Grimme from 2014 published in WIREs 4, 576 (2014). Note that this is already 7 years old; many double hybrids have appeared since then. However, it should give you a good overview.
Susi is right that these are called double hybrids, I'll just add a bit more to complement that correct answer. My answer to "What are some recent developments in density functional theory?" was about double hybrids, and it looks like the community here considered that to be one of the most important "recent" developments in DFT.
About the MP2 contributions, single contributions are ignored (this is already the case when starting from Hartree-Fock because Brillouin's theorem says they don't exist, but when starting from a DFT calculation single excitations can contribute, and early developers of double hybrid functionals decided not to include them), see this: Why are single excitations ignored in the MP2 component of double hybrid functional calculations?
Another interesting discussion about double hybrids can be found here: Physically motivated double hybrid DFT?, and here: Using double-hybrid functional instead of MP2 to avoid basis set dependence, and if you're interested in going even further than MP2, there's been some papers that used QCISD or RPA: Extended Hybrid Methods.
Finally, double hybrids have been recommended or their accuracy has been discussed, in a lot of answers on this site, for example: