Let $W$ be an affine scheme and let $f : W \to X$ be a surjective \'etale morphism. There exists an integer $d$ such that all geometric fibres of f have $\leq d$ points (because $X$ is a separated algebraic hence reasonable, see Decent Spaces, Lemma \ref{decent-spaces-lemma-bounded-fibres}). Picking $d$ minimal we get a nonempty open $U \subset X$ such that $f^{-1}(U) \to U$ is finite \'etale of degree $d$, see Decent Spaces, Lemma \ref{decent-spaces-lemma-quasi-compact-reasonable-stratification}. Let $$ V \subset W \times_X W \times_X \ldots \times_X W $$ ($d$ factors in the fibre product) be the complement of all the diagonals. Because $W \to X$ is separated the diagonal $W \to W \times_X W$ is a closed immersion. Since $W \to X$ is \'etale the diagonal $W \to W \times_X W$ is an open immersion, see Morphisms of Spaces, Lemmas \ref{spaces-morphisms-lemma-etale-unramified} and \ref{spaces-morphisms-lemma-diagonal-unramified-morphism}. Hence the diagonals are open and closed subschemes of the quasi-compact scheme $W \times_X \ldots \times_X W$. In particular we conclude $V$ is a quasi-compact scheme. Choose an open immersion $W \subset Y$ with $Y$ H-projective over $A$ (this is possible as $W$ is affine and of finite type over $A$; for example we can use Morphisms, Lemmas \ref{morphisms-lemma-quasi-affine-finite-type-over-S} and \ref{morphisms-lemma-H-quasi-projective-open-H-projective}). Let $$ Z \subset Y \times_A Y \times_A \ldots \times_A Y $$ be the scheme theoretic image of the composition $V \to W \times_X \ldots \times_X W \to Y \times_A \ldots \times_A Y$. Observe that this morphism is quasi-compact since $V$ is quasi-compact and $Y \times_A \ldots \times_A Y$ is separated. Note that $V \to Z$ is an open immersion as $V \to Y \times_A \ldots \times_A Y$ is an immersion, see Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-immersion}. The projection morphisms give $d$ morphisms $g_i : Z \to Y$. These morphisms $g_i$ are projective as $Y$ is projective over $A$, see material in Morphisms, Section \ref{morphisms-section-projective}. We set $$ X' = \bigcup g_i^{-1}(W) \subset Z $$ There is a morphism $X' \to X$ whose restriction to $g_i^{-1}(W)$ is the composition $g_i^{-1}(W) \to W \to X$. Namely, these morphisms agree over $V$ hence agree over $g_i^{-1}(W) \cap g_j^{-1}(W)$ by Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-equality-of-morphisms}. Claim: the morphism $X' \to X$ is proper. \medskip\noindent If the claim holds, then the lemma follows by induction on $d$. Namely, by construction $X'$ is H-quasi-projective over $\Spec(A)$. The image of $X' \to X$ contains the open $U$ as $V$ surjects onto $U$. Denote $T$ the reduced induced algebraic space structure on $X \setminus U$. Then $T \times_X W$ is a closed subscheme of $W$, hence affine. Moreover, the morphism $T \times_X W \to T$ is \'etale and every geometric fibre has $< d$ points. By induction hypothesis there exists a proper surjective morphism $T' \to T$ where $T'$ is a scheme H-quasi-projective over $\Spec(A)$. Since $T$ is a closed subspace of $X$ we see that $T' \to X$ is a proper morphism. Thus the lemma follows by taking the proper surjective morphism $X' \amalg T' \to X$. \medskip\noindent Proof of the claim. By construction the morphism $X' \to X$ is separated and of finite type. We will check conditions (1) -- (4) of Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-refined-valuative-criterion-universally-closed} for the morphisms $V \to X'$ and $X' \to X$. Conditions (1) and (2) we have seen above. Condition (3) holds as $X' \to X$ is separated (as a morphism whose source is a separated algebraic space). Thus it suffices to check liftability to $X'$ for diagrams $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] & V \ar[d] \\ \Spec(R) \ar[r] & X } $$ where $R$ is a valuation ring with fraction field $K$. Note that the top horizontal map is given by $d$ pairwise distinct $K$-valued points $w_1, \ldots, w_d$ of $W$. In fact, this is a complete set of inverse images of the point $x \in X(K)$ coming from the diagram. Since $W \to X$ is surjective, we can, after possibly replacing $R$ by an extension of valuation rings, lift the morphism $\Spec(R) \to X$ to a morphism $w : \Spec(R) \to W$, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-lift-valuation-ring-through-flat-morphism}. Since $w_1, \ldots, w_d$ is a complete collection of inverse images of $x$ we see that $w|_{\Spec(K)}$ is equal to one of them, say $w_i$. Thus we see that we get a commutative diagram $$ \xymatrix{ \Spec(K) \ar[r] \ar[d] &  Z \ar[d]_{g_i}\\ \Spec(R) \ar[r]^w & Y } $$ By the valuative criterion of properness for the projective morphism $g_i$ we can lift $w$ to $z : \Spec(R) \to Z$, see Morphisms, Lemma \ref{morphisms-lemma-locally-projective-proper} and Schemes, Proposition \ref{schemes-proposition-characterize-universally-closed}. The image of $z$ is in $g_i^{-1}(W) \subset X'$ and the proof is complete.