Inspired by the discovery of a doubly charmed tetraquark state $T_{cc}^+$ by the LHCb Collaboration and based on the prediction of a doubly charmed $DD^{ast}$ molecule with $I(J^{P})=0(1^{+})$ in our recent works, [Phys.Rev.D 102 (2020), 091502] and [Phys.Rev.D 99 (2019), 094018], we employ the effective Lagrangian approach to investigate the decay width of $T_{cc}to D Dpi$ and $T_{cc}to DDgamma$. We show that both the $T_{cc}to D Dpi$ and $T_{cc}to DDgamma$ modes contribute to the decay width of $T_{cc}$, with the former playing a more important role.However, within the $DD^*$ molecule picture, the obtained decay width is rather small compared with the experimental value of $Gamma=410pm175$ keV. We argue that the existence of a compact tetraquark component cannot lead to an appreciable increase of the decay width. As a result, the discrepancy may indicate that either $T_{cc}$ is more likely a near threshold $DD^*$ resonance or the decay width is close to the lower experimental boundary.
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