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For del-Pezzo surfaces (except possibly $\mathbb{P}^1 \times \mathbb{P}^1$) the answer is yes. There is a $K3$ which is a branched double cover of $\mathbb{P}^{2}$ blown up in $9$ points (example 1.2.ii)) here http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf). Hence we get a finite morphism to any del-Pezzo which is a blow-up of $\mathbb{P}^{2}$.