Comment to Sebastian answer
If the stable vector bundle $E$ be of negative degree then there is no non-zero holomorphic sections due to Kobayashi and also Wu. A non-zero holomorphic section of a vector bundle $E$ on $X$ gives an injective homomorphism from $\mathcal O_X$ to $E$. So, the image of this homomorphism is of degree zero but the stable vector bundle $E$ has negative degree.
Line bundles with zero degree admit non non-zero holomorphic section.
IFrom Kobayshi In general for stable vector bundles with zero degree all the holomorphic sections are parallel
About your question I don't think such moduli space be smooth in general.
Let $X$ be a complex affine scheme, it is said to have quadratic algebraic singularities if it is defined by finitely many quadratic homogeneous polynomials. Let $X$ be an analytic space, it is said to have quadratic algebraic singularities if it is locally isomorphic to complex affine schemes with quadratic algebraic singularities.
We know from the following work of NadelNadel
Nadel, Alan Michael Singularities and Kodaira dimension of the moduli space of flat Hermitian-Yang-Mills connections Compositio Mathematica, Volume 67 (1988) no. 2 , p. 121-128
moduli space of flat locally free sheaves on a compact complex Kahler manifold which are stable have quadratic algebraic singularities.
Nadel showed that the Moduli space of Hermitian-Einstein metric on stable vector bundles have quadratic algebraic singularites,
So I guess you must work on such type singularites for your moduli space of Higgs bundles with a section