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?