For measurable cardinals, the answer is yes and is due to Sy Friedman. See Coding Over a Measurable Cardinal.
There is some difficulty to extend the result to the context of Woodin cardinals, see Genericiy and large cardinals.
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.