Recently, Liu et al. proposed a so-called extended Anderson-Higgs mechanism by studying the (2+1)-dimensional Ginzburg-Landau model in the pseudogap region of high-T_c superconductor (Phys. Rev. B 65 (2002) 132513). We revisit this problem based on a general decomposition of the U(1) gauge potential. Using the bulk superconductor and superconduct ring as examples, we obtain a simpler expression for the extended Anderson-Higgs mechanism. In the former case we indicate that all the phase field can always be "eaten up'' by the pure gauge term A_||. In the latter case, we decompose the phase field as θ(x)=θ₁(x)+θ₂(x) and find that only the phase field θ₁ connected with Anderson-Higgs mechanism can be canceled by the pure-gauge term A_||. On the other hand, the remaining phase field θ₂ connected with A_^is multi-valued, which can induce new physical effects such as A-B effect and flux quantization. It is natural to conclude that there is no longitudinal phase fluctuation effect in high-temperature superconductors since longitudinal phase \theta₁ is connected with pure-gauge term.