We investigated a one-dimensional system of anyons that interact with each other under a local three-body term. Using a fractional Jordan-Wigner transformation, we arrived at a modified Bose-Hubbard model, which exhibits gapped and gapless phases. We built the phase diagram of the system fixing the hopping parameter or the statistics, showing the evolution of the critical points, which were estimated with von Neumann block entropy. A superfluid to Mott insulator quantum phase transition with one particle per site can be driven by the statistics or the interaction. Specifically, we show that for larger angles there is a finite critical value of the interaction at which the Mott phase appears. Also, we found that the critical angles increase with the hopping. Diverse gapless phases were observed away from the pseudofermion limit.