Let [Formula: see text] and [Formula: see text] be non-zero integers with [Formula: see text]. An integer is called [Formula: see text]-free if it is not divisible by the [Formula: see text]th power of a prime. A result of Mirsky states that there are infinitely many primes [Formula: see text] such that [Formula: see text] is [Formula: see text]-free. In this paper, we study an additive Goldbach-type problem and prove two uniform distribution results using these primes. We also study certain properties of primes [Formula: see text] such that [Formula: see text] are simultaneously [Formula: see text]-free, where [Formula: see text] are non-zero integers and [Formula: see text].