I'm reading the Atanas Atanasov's course notes of Joe Harris' course Geometry of Algebraic Curves and have a question about modification of an dimension counting argument applying deformation theory.

On page 22 one consideres a version of Hurwitz scheme

$$ V_{d,g}:= \{(X, f: X \to \mathbb{P}^2) \ \vert \ X \text{ curve of genus } g, f \text{ has degree } d \text{ and is birational } \\ \text{ onto a plane curve with } \delta \text{ nodes } \} $$

together with two canonical canonical projections $V_{d,g} \to M_g $ (to the 'naive' moduli set) and $V_{d,g} \to \mathbb{P}^{\delta} \backslash \Delta$.

Rather elementary considerations in the script show that $\dim V_{d,g}= 3d+g-1$ if $d(d+3)/2 \ge 3 \delta$ but the Remark 4.2 says:

There is a serious problem with this argument if $3 \delta> d(d + 3)/2 $ but this can be fixed using deformation theory.

Could somebody elaborate this deformation theoretic argument fixing the gap the remark 4.2 is refering to?