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Let's define the following:

(Def) A continuous map $f_0 : A \to B$ where $A,B$ are topological spaces is called essential if every homotopic map $f_1$ is surjective, i.e. $f_1 (A) = B$.

Let $B=S^n$ be the $n$-sphere and let $f_0 : A \to B$ be non-essential. Claim: Then there exists a homotopic map $f_1$ such that $f_1 (A) = \ast$ is a one point space.

That's a footnote in the paper I'm reading. But I don't understand the claim. If $A$ is simply-connected then I think I "see" that $f_0(A) = f_1(A) = \ast$. But the claim in the paper does not put any restrictions on $A$ so in particular, it could be disconnected. I'm a bit confused. Would someone show me how the claim is true? Thanks lots.

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You confused homotopic maps with homotopic spaces, as pointed out by DrKW in the comments. Here is what you can do to the image if $f$ is not surjective:

This does not work if $f$ is surjective since then you'd have to "tear" the image. But of course something that "tears" your space cannot be continuous.

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