We consider a non-linear PDE on R^d with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity is of quadratic type in the gradient of the unknown. Under suitable conditions on the parameters we prove local existence and uniqueness of a mild solution to the PDE, and investigate properties like continuity with respect to the initial condition. To conclude, we consider an application of the PDE to stochastic analysis, in particular to a class of non-linear backward stochastic differential equations with distributional drivers.