Cut and polish to try to take the cruddy layer off, and then apply a coat of wax to protect what's then left exposed.

It could stay good for a week or 3 months. Results do vary.

I had a similar problem with the bonnet on my WRX and did the above with the cut and polish and then wax. It would last about 2 weeks at a time before it would turn cruddy/milky again. In the end I went and got it resprayed. It's really not as expensive as you think ($400 for bonnet in my case, and I went and had other things sprayed too and got it all done in one hit). It was drive in and drive out. Simple.