The main result is an improvement of previous results on the equation
\[
f
(
x
)
+
f
(
y
)
−
f
(
x
+
y
)
=
g
[
ϕ
(
x
)
+
ϕ
(
y
)
−
ϕ
(
x
+
y
)
]
f(x)+f(y)-f(x+y)=g[\phi (x)+\phi (y)-\phi (x+y)]
\]
for a given function
ϕ
\phi
. We find its general solution assuming only continuous differentiability and local nonlinearity of
ϕ
\phi
. We also provide new results about the more general equation
\[
f
(
x
)
+
f
(
y
)
−
f
(
x
+
y
)
=
g
(
H
(
x
,
y
)
)
f(x)+f(y)-f(x+y)=g(H(x,y))
\]
for a given function
H
H
. Previous uniqueness results required strong regularity assumptions on a particular solution
f
0
,
g
0
f_{0},g_{0}
. Here we weaken the assumptions on
f
0
,
g
0
f_{0},g_{0}
considerably and find all solutions under slightly stronger regularity assumptions on
H
H
.