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In the thermodynamics book (Adkins) I'm using, the following relation is cited without reference but I am not sure where it comes from.

$$\left(\frac{\partial S}{\partial T}\right)_P=\left(\frac{\partial S}{\partial T}\right)_V+\left(\frac{\partial S}{\partial V}\right)_T\left(\frac{\partial V}{\partial T}\right)_P$$

I am not sure what combination of mathematical and physical relations leads to this but any help would be appreciated!

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Here, $S$ is a function of $V$ and $T$. Here, we want to know $ \left (\frac{\partial S(V,T)}{\partial T} \right )_P$.
By chain rule, $\left (\frac{\partial S(V,T)}{\partial T} \right )_P = \left (\frac{\partial S(V,T)}{\partial T} \right )_V\times \left(\frac{\partial T }{\partial T}\right)_P + \left (\frac{\partial S(V,T)}{\partial V} \right )_T\times \left (\frac{\partial V }{\partial T} \right )_P$.
When you simplify this, you get your equation.

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