We consider a general branching population where the lifetimes of individuals are i.i.d. with arbitrary distribution and each individual gives birth to new individuals at Poisson times independently from each others. The population counting process is a Binary Homogeneous Crump-Jargers-Mode process. Moreover, we suppose that the individuals experience mutations at Poisson rate θ under the infinite-alleles assumption. These mutations lead to a partition of the population by type. In this paper, we obtain central limit theorems for the frequency spectrum when time goes to infinity. We also obtain CLT for the population counting process. To this end, we use classical renewal theory and recent works [18, 6, 5] on this model to obtain the moments of the error. Then, we can precisely study the asymptotic behaviour of these moments thanks to Lévy processes theory. These results in conjunction with a new decomposition of the splitting tree allow us to obtain the wanted Theorems.