The BESIII Collaboration recently reported the observation of the $a_0(980)^0-f_0(980)$ mixing in the isospin breaking decay $J/psito etapi^0phi$. In the Dalitz plot for that decay with the $eta$ reconstructed from two photons, there is a band around $1.4$~GeV on the $pi^0 phi$ distribution. In general, this peak can be due to a resonance or a kinematic effect. In this paper, we study the effects of a set of $K^*Kbar K$ triangle diagrams, and show that due to triangle singularities such diagrams can lead to a peak around 1.4~GeV in the $pi^0phi$ invariant mass distribution. The Dalitz plot induced by such a mechanism has a feature consistent with the BESIII observation, namely events along the band accumulate at both ends close to the Dalitz plot boundary. The effect of the same mechanism on the $J/psito pi^0pi^0phi$ and $J/psi to etapi^0 K^+K^-$ decays are also investigated. We suggest to take more data for the $J/psito etapi^0phito etapi^0 K^+ K^-$ and check whether the structure around $1.4$~GeV persists for the $K^+K^-$ invariant mass away from the $phi$ mass region. This is crucial for understanding whether the band is due to triangle singularities or due to a resonance. Were it the latter, the band should remain while it would not if it is due to the former.
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