We　propose　a　Lawson　-time　-splitting　extended　Fourier　pseudospectral　(LTSeFP)　method　for　the　numerical　integration　of　the　Gross-Pitaevskii　equation　with　time　-dependent　potential　that　is　of　low　regularity　in　space.　For　the　spatial　discretization　of　low　regularity　potential,　we　use　an　extended　Fourier　pseudospectral　(eFP)　method,　i.e.,　we　compute　the　discrete　Fourier　transform　of　the　low　regularity　potential　in　an　extended　window.　For　the　temporal　discretization,　to　efficiently　implement　the　eFP　method　for　time　-dependent　low　regularity　potential,　we　combine　the　standard　time　-splitting　method　with　a　Lawson　-type　exponential　integrator　to　integrate　potential　and　nonlinearity　differently.　The　LTSeFP　method　is　both　accurate　and　efficient:　it　achieves　first　-order　convergence　in　time　and　optimal　-order　convergence　in　space　in　L　2　-norm　under　low　regularity　potential,　while　the　computational　cost　is　comparable　to　the　standard　time　-splitting　Fourier　pseudospectral　method.　Theoretically,　we　also　prove　such　convergence　orders　for　a　large　class　of　spatially　low　regularity　time　-dependent　potential.　Extensive　numerical　results　are　reported　to　confirm　the　error　estimates　and　to　demonstrate　the　superiority　of　our　method.