Fix local coordinates $x^i$ on $M$ and define the usual coordinates $(x^i, \xi^i = dx^i)$ on $TM$. The vertical bundle is then just the span of $\partial/\partial \xi^i$, while the horizontal space at $v \in T_p M$ can be written as the span of the vectors $\def\pfrac#1#2{\frac{\partial #1}{\partial #2}}$ $$h_i=\pfrac{}{x^i} -
\Gamma_i^j(v)\pfrac{}{\xi^j}$$ for some coefficients $\Gamma_i^j(v)$. You can check that we then have the formula $$\nabla_X Y = X^i \left(\pfrac{Y^j}{x^i}+\Gamma^j_i(Y)\right)\pfrac{}{x^j}\tag{1}$$ for the covariant derivative. When $\Gamma^j_i$ is linear (i.e. $\Gamma^j_i(Y) = \Gamma^j_{ik} Y^k$) this is exactly the usual coordinate formula for an affine connection with Christoffel symbols $\Gamma^j_{ik}.$ The assumption that $H$ is $m_\lambda$-invariant is not quite as strong as linearity, but it does imply positive homogeneity: in coordinates we have $m_\lambda(x,\xi) = (x, \lambda \xi)$; so $Dm_\lambda(\partial/\partial x^i) = \partial/\partial x^i$ and $Dm_\lambda(\partial/\partial \xi^i) = \lambda \partial/\partial \xi^i$ and thus $$Dm_\lambda(h_i|_v) = \pfrac{}{x^i}\Big|_{\lambda v} - \lambda \Gamma^j_i(v) \pfrac{}{\xi^j}\Big|_{\lambda v}$$which can only equal $h_i|_{\lambda v}$ if we have $\lambda \Gamma^j_i(v) = \Gamma^j_i(\lambda v).$
(edit: actually, homogeneity is just as strong as linearity if you're assuming differentiability - see here.)
We now have everything we need to prove the Leibniz rule: applying $(1)$ to both $Y$ and $fY$ we find
$$\begin{align}
(\nabla_i Y)^j &= \pfrac{Y^j}{x^i}+\Gamma^j_i(Y) \\
(\nabla_i (fY))^j &= f \pfrac{Y^j}{x^i} + \pfrac{f}{x^i} Y^j + \Gamma^j_i(f Y).
\end{align}$$
Applying the homogeneity of $\Gamma$ to the last term and multiplying the first equation by $f$ we see $$(\nabla_i(fY))^j = f(\nabla_i Y)^j + \pfrac{f}{x^i} Y^j$$ as desired.
