Defintion of VNP

A family of polynomials $${f_n}$$ over $$mathbb{F}$$ is $$p$$-definable if there exists two polynomially bounded functions $$t,kcolon mathbb{N} longrightarrow mathbb{N}$$ and a family $${g_n}$$ in $$mathsf{VP}_{mathbb{F}}$$ such that for every $$n$$

$$f_n(x_1,ldots,x_{k(n)}) = sum_{w in {0,1}^{t(n)}}g_{t(n)}(x_1,ldots,x_{k(n)},w_1,ldots,w_{t(n)})$$

This is a definition of $$mathsf{VNP}_{mathbb{F}}$$.

I am trying to understand this definition. To me it appears that a polynomial $$f$$ is in the class VNP if there it can be computed as a sum of other polynomials which admits a circuit of size polynomial size. How to read the definition of VNP like the way I am trying to understand it?