Retraction onto a circle in a simplicial complex
As mentioned in Cihan's answer, this is a result of Borsuk. The proof is divided into two papers (see [1, Théorème 30] and [2, Korollar 11]). A proof of this result (for Peano continua and rather difficult to read) appears in Kuratowski's book [3, S 57, III, Theorem 4].
In Section 1.1.3 (pages 8-11) of my thesis [4], I provide a proof for arbitrary (not necessarily finite) simplicial complexes which is based on Kuratowski's proof.
I sketch here a different way to prove this result for a finite simplicial complex $X$. First we take a 1-cycle in $Z_1(X)$ which represents a generator of a direct summand of $H_1(X)$ isomorphic to $\mathbb{Z}$. Then we need to take a representative of this cycle which is a simple closed curve. If $X$ is not a surface and $X$ has no local separating points (a point $x\in X$ is a local separating point if there is a connected open neighborhood $U\ni x$ such that $U-x$ is disconnected) it is possible to do this. The main idea is to consider three 2-simplices with a common edge and to use this to remove intersections one by one. It is easy to reduce to the case in which $X$ has no local separating points. Finally, surfaces are managed using the classification.
In every proof I know, it is necessary to subdivide the complex at some point. I do not know if it is always possible to find a subcomplex of the original complex homeomorphic to $S^1$ which is a retract.
[1] K. Borsuk. Quelques théorèmes sur les ensembles unicohérents, Fund. Math. 17 (1931), no. 1, 171–209.
[2] K. Borsuk. Über die Abbildungen der metrischen kompakten Räume auf die Kreislinie, Fund. Math. 20 (1933), no. 1, 224–231.
[3] K. Kuratowski. Topology vol. ii, Academic Press, 1968.
[4] I. Sadofschi Costa. Fixed points of maps and actions on 2-complexes, PhD thesis, Universidad de Buenos Aires, 2019. Available at http://cms.dm.uba.ar/academico/carreras/doctorado/tesisSadofschi.pdf.
Interesting question. I believe it is sufficient, let me sketch a proof.
Since $$b_1(X) = \text{rank}(H_1(X;\mathbb{Z})) = \text{rank}(H^1(X;\mathbb{Z})) $$ it follows from $b_1(X) \ne 0$ that $H^1(X;\mathbb{Z})$ is nontrivial. Consider the canonical bijection $[X,S^1] \approx H^1(X;\mathbb{Z})$, which associates to each $f : X \to S^1$ the pullback via $f$ of the fundamental cohomology class of $S^1$, denoted $f^*(d\theta) \in H^1(X;\mathbb{Z})$ (if I may abuse notation). Let $f : X \to S^1$ be any element whose pullback class $f^*(d\theta)$ is a basis element of $H^1(X;\mathbb{Z})$. It follows that there exists a continuous map $\sigma : S^1 \to X$ such that the $f^*(d\theta)$ evaluates to $+1$ on $\sigma$, and so $f \circ \sigma : S^1 \to S^1$ is homotopic to the identity.
Claim: One can homotope $f$ and $\sigma$ so that $f \circ \sigma$ is equal to to the identity on $S^1$.
Once this claim is proved, it follows that $\sigma \circ f : X \to \sigma(S^1)$ is a retract onto a circle.
Let me sketch a proof of the claim. First, by homotoping $\sigma$, we may assume that $\sigma$ is a concatenation of 1-cells of $X$, $\sigma = e_1 \cdots e_K$. Next, we may assume that for each 1-cell $e_k$, the restriction $f | e_k$ goes around $S^1$ monotonically, either forward, backward, or constant; this is true by the homotopy extension lemma. And then we can assume that $f$ is not constant on each $e_k$, or else we may do a small homotopy of $f$ near an endpoint of $e_k$, and then homotope $f$ to be non constant on $e_k$, and then apply homotopy extension.
Let $P$ be the number of local maxima of $f \circ \sigma$, which is equal to the number of local minima. The proof now proceeds by induction on $P$. If $P=1$ it should be clear that $f \circ \sigma$ is a homeomorphism, and then by composing with an isotopy of $\sigma$ we can make $f \circ \sigma$ the identity. If $P \ge 2$ then (after a cyclic permutation) we may assume that the restrictions of $\sigma$ to $e_K$ wraps negatively, to $e_1,...,e_J$ wrap positively, and to $e_{J+1}$ wraps negatively. Now we do a subsidiary induction on $J$: to reduce $J$, homotope $\sigma \mid e_{J-1} e_J$ to be monotonic, and apply homotopy extension.