For measurable cardinals, the answer is yes and is due to Sy Friedman. See  [Coding Over a Measurable Cardinal](https://www.jstor.org/stable/2274810?origin=crossref&seq=1). 

There is some difficulty to extend the result to the context of Woodin cardinals, see [Genericiy and large cardinals](https://www.worldscientific.com/doi/abs/10.1142/S0219061305000420).

For strong cardinals, this is open, though for some simple sets, say the collection of Prikry sequences for the Prikry product, the answer is yes, see [Killing the GCH everywhere with a single real](https://projecteuclid.org/journals/journal-of-symbolic-logic/volume-78/issue-3/Killing-the-GCH-everywhere-with-a-single-real/10.2178/jsl.7803060.short).

See also [Coding over core models](http://www.math.uni-bonn.de/people/david/files/fri-schi-schr-coding-K.pdf)