Strong covering without squares

Let W be an inner model of ZFC. Let κ be a cardinal in V. We say that κ-covering holds between V and W iff for all X ∈ V with X ⊆ ON and V ⊨ |X| < κ, there exists Y ∈ W such that X ⊆ Y ⊆ ON and V ⊨ |Y| < κ. Strong κ-covering holds between V and W iff for every structure M ∈ V for some countable first-order language whose underlying set is some ordinal λ, and every X ∈ V with X ⊆ λ and V ⊨ |X| < κ, there is Y ∈ W such that X ⊆ Y ≺ M and V ⊨ |Y| < κ.   We prove that if κ is V-regular, ${\displaystyle {\kappa }^{+}\_V={\kappa }^{+}\_W}$, and we have both κ-covering and ${\displaystyle {\kappa }^{+}}$-covering between W and V, then strong κ-covering holds. Next we show that we can drop the assumption of ${\displaystyle {\kappa }^{+}}$-covering at the expense of assuming some more absoluteness of cardinals and cofinalities between W and V, and that we can drop the assumption that ${\displaystyle {\kappa }^{+}\_W={\kappa }^{+}\_V}$ and weaken the ${\displaystyle {\kappa }^{+}}$-covering assumption at the expense of assuming some structural facts about W (the existence of certain square sequences).