Let \(r_1, \dots , r_t\) be positive integers with \(r_1^{-1}+\cdots +r_t^{-1}\ge 1.\) Recently, Chen and Xu proved that the set of positive integers that can be written as \(p+2^{k_1^{r_1}}+\cdots +2^{k_t^{r_t}},\) where \(k_1, \dots , k_t \) are positive integers and p is prime, has a positive lower asymptotic density. In this paper, we generalize their result. In particular, let \(\mathcal {L}=\{L_n: n=0,1,2,\ldots \} \) be the Lucas sequence. We prove that the set of positive integers that can be written as \(p+L_{ k_1^{r_1} }+\cdots +L_{k_t^{r_t} }\) has a positive lower asymptotic density. For \( t=r_1=1, \) we show that there is a positive proportion of all positive integers that can be uniquely represented as the sum of a prime and a Lucas number.
