You cannot get a single-input, single-output transfer function of this system that takes both \$u\$ and \$c\$ into account.
Laplace-domain analysis only works for linear systems, i.e. ones that obey superposition. Strictly speaking, adding an independent constant makes the system nonlinear (you can check me on that: \$y = a x\$ is linear, and obeys superposition, but \$y = a x + b\$ does not). The resulting system is affine, which is the world's easiest nonlinearity to resolve, but it's still nonlinear.
You must do one of two things: do your linear analysis while ignoring \$c\$ then patch up the results, or treat the system as a multi-input, single-output system.
To "ignore" \$c\$, set \$c = 0\$, do your analysis, then add it back to the output.
There's a number of ways to treat the system as MISO. The easiest is to find two transfer functions: \$\frac{V(s)}{U(s)}\$ and \$\frac{V(s)}{C(s)}\$. Then -- because you've modeled a system that's linear but has two inputs -- you can use superposition to find \$v\$ by adding the system response to \$u\$ and the system response to \$c\$.
Alternately, you could model your system in state space, or you could model it with a vector-valued transfer function; i.e. $$H(s) = \begin{bmatrix}\frac{V(s)}{U(s)} & \frac{V(s)}{C(s)}\end{bmatrix}.$$
