The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an $L^2$–$L^p$ restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime $k$-tuples. Let $a_1,\cdots ,a_k$ and $b_1,\cdots ,b_k$ be positive integers. Write $h\left(\theta \right):={\sum }_{n\in X}e\left(n\theta \right)$, where $X$ is the set of all $n\le N$ such that the numbers $a_{1}n+b_1,\cdots ,a_{k}n+b_k$ are all prime. We obtain upper bounds for ${\parallel h\parallel }_{L^p\left(𝕋\right)}$, $p\>2$, which are (conditionally on the Hardy-Littlewood prime tuple conjecture) of the correct order...