In "$L_1$-Norms of Exponential Sums and the Corresponding Additive Problem" Garaev and Kueh prove that if $F$ is in $C^3([1,N])$ with $F'(x)>0$, $F''(x)>0$ and $F'''(x)<0$, then the additive energy of $A=\{\lfloor F(1)\rfloor,\ldots,\lfloor F(N)\rfloor\}$ satisfies $$ \frac{N^4}{F(N)+1}\ll E(A)\ll (F'(1)+1)N^{5/2} +\frac{N^2\log N}{F''(N)}.$$
In particular, if $F(x)=Ax^c$, with $A>0$ and $1< c\leq 3/2$, then $$ N^{4-c}\ll_{c,A} E(A)\ll_{c,A} N^{4-c}\log N.$$
This paper builds on earlier work of Garaev, which presumably uses indicator trick mentioned in the first answer to bound the exponential sums, however the exponential sum estimates do not appear explicitly in the Garaev-Kueh paper.
PS: The ISSN for the Garaev-Kueh paper is 0232-2064.
PPS: I think the method mentioned by Eric might work for $1<c<2$, but I didn't write down all the details. You should be able to figure it out by staring at Heath-Brown's paper for a while (you should be able to avoid eq (5) if you're not summing over primes; proving (5) occupies the next 20 pages; in particular, you can re-apply Lemma 1 to the exponential sum at the end of the chain of inequalities on p. 247; overall I think the error term is $O(N^{1/2}(\log N)^b)$ for some power $b$.