When dealing with thermodynamic equalities, it is very useful to introduce Jacobian determinants $ \frac{\partial (u,v)}{\partial (x,y)} = \begin{vmatrix} (\partial_x u)_y & (\partial_y u)_x \\ (\partial_x v)_y & (\partial_y v)_x \end{vmatrix}.$ Here, we have and will implicitly assume that $u(x,y)$ and $v(x,y)$. But because we often change the variables which the function depends on, we always keep the variable which is kept constant as a subscript to the bracket surrounding the partial derivative. In thermodynamics, you should always think that functions are defined implicitly. All quantities live on a 2D surface. Thus specifying two independent coordinates you can figure out a third one.
The Jacobian have the following relevant properties: $\frac{\partial (u,y)}{\partial (x,y)} = \left( \frac{\partial u}{\partial x} \right)_y,$ $\frac{\partial (u,v)}{\partial (x,y)} = \left(\frac{\partial (x,y)}{\partial (u,v)} \right)^{-1}, \qquad\text{(inverse function theorem)}$ and $\frac{\partial (u,v)}{\partial (x,y)} = \frac{\partial (u,v)}{\partial (s,t)} \frac{\partial (s,t)}{\partial (x,y)} . \qquad\text{(chain rule)}$
In your case $\left(\frac{\partial U}{\partial T} \right)_p= \frac{\partial (U,p)}{\partial (T,p)} = \frac{\partial(U,p)/\partial(T,V) }{\partial(T,p)/\partial(T,V)} = \frac{(\partial U/\partial T)_V (\partial p/\partial V)_T - (\partial U/\partial V)_T (\partial p/\partial T)_V}{\left(\partial p/\partial V\right)_T}. $ Now, we evaluate $\frac{(\partial p/\partial T)_V}{(\partial p/\partial V)_T} = \frac{\partial(p,V)}{\partial(T,V)} \Big/\frac{\partial(p,T)}{\partial(V,T)} = - \frac{\partial(p,V)}{\partial(V,T)} \frac{\partial(V,T)}{\partial(p,T)} = - \frac{\partial(p,V)}{\partial(p,T)} = -\left(\frac{\partial V}{\partial T}\right)_p .$ So in total, we have $\left(\frac{\partial U}{\partial T} \right)_p =\left(\frac{\partial U}{\partial T}\right)_V +\left( \frac{\partial U}{\partial V}\right)_T \left(\frac{\partial V}{\partial T}\right)_p. $
Note that we did not use any Maxwell relation. Your relation is just a fact about partial derivatives.