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We prove that for sets $A, B, C \subset \mathbb{F}_p$ with $|A|=|B|=|C| \leq\sqrt{p}$ and a fixed $0 \neq d \in \mathbb{F}_p$ holds  $$  \max(|AB|, |(A+d)C|) \gg|A|^{1+1/26}.  $$  In particular,  $$  |A(A+1)| \gg |A|^{1 + 1/26}  $$  and  $$  \max(|AA|, |(A+1)(A+1)|) \gg |A|^{1 + 1/26}.  $$  The first estimate improves the bound by Roche-Newton and Jones.  In the general case of a field of order $q = p^m$ we obtain similar estimateswith the exponent $1+1/559 + o(1)$ under the condition that $AB$ does not havelarge intersection with any subfield coset, answering a question ofShparlinski.  Finally, we prove the estimate  $$  \left| \sum_{x \in \mathbb{F}_q} \psi(x^n) \right| \ll q^{\frac{7 -2\delta_2}{8}}n^{\frac{2+2\delta_2}{8}}  $$  for Gauss sums over $\mathbb{F}_q$, where $\psi$ is a non-trivial additivecharacter and $\delta_2 = 1/56 + o(1)$. The estimate gives an improvement overthe classical Weil bound when $q^{1/2} \ll n = o\left( q^{29/57 + o(1)}\right)$.

On additive shifts of multiplicative almost-subgroups in finite fields