Generalized Cauchy difference equations. II

The main result is an improvement of previous results on the equation

$$f(x)+f(y)-f(x+y)=g[\phi(x)+\phi(y)-\phi(x+y)]$$

for a given function $\phi$. We find its general solution assuming only continuous differentiability and local nonlinearity of $\phi$. We also provide new results about the more general equation

$$f(x)+f(y)-f(x+y)=g(H(x,y))$$

for a given function $H$. Previous uniqueness results required strong regularity assumptions on a particular solution $f_{0},g_{0}$. Here we weaken the assumptions on $f_{0},g_{0}$ considerably and find all solutions under slightly stronger regularity assumptions on $H$.