We present a calculation of the pion quark momentum fraction, ⟨x⟩, and its third Mellin moment ⟨x2⟩. We also obtain directly, for the first time, ⟨x⟩ and ⟨x2⟩ for the kaon using local operators. We use an ensemble of two degenerate light, a strange, and a charm quark (Nf=2+1+1) of maximally twisted mass fermions with clover improvement. The quark masses are chosen so that they reproduce a pion mass of about 260 MeV and a kaon mass of 530 MeV. The lattice spacing of the ensemble is 0.093 fm and the lattice has a spatial extent of 3 fm. We analyze several values of the source-sink time separation within the range of 1.12–2.23 fm to study and eliminate excited-state contributions. The necessary renormalization functions are calculated nonperturbatively in the RI′ scheme and are converted to the modified minimal subtraction scheme at a scale of 2 GeV. The final values for the momentum fraction are ⟨x⟩u+π=0.261(3)stat(6)syst, ⟨x⟩u+K=0.246(2)stat(2)syst, and ⟨x⟩s+K=0.317(2)stat(1)syst. For the third Mellin moments, we find ⟨x2⟩u+π=0.082(21)stat(17)syst, ⟨x2⟩u+K=0.093(5)stat(3)syst, and ⟨x2⟩s+K=0.134(5)stat(2)syst. The reported systematic uncertainties are due to excited-state contamination. We also give the ratio ⟨x2⟩/⟨x⟩ which is an indication of how quickly the parton distribution functions lose support at large x.