Inelastic neutron scattering has been used to study the in-plane Cu-O bond-stretching mode in oxygen doped La$_{1.94}$Sr$_{0.06}$CuO$_{4.035}$ ($T_c = 38,text{K}$) and La$_2$CuO$_{4+delta}$ ($T_c = 43,text{K}$). Similar to results from optimally doped La$_{1.85}$Sr$_{0.15}$CuO$_4$ ($T_c = 35,text{K}$), we observe anomalous features in the dispersion of this half-breathing mode in the form of a softening halfway through the Brillouin Zone. Considering the differences in electronic structure and local environment between the oxygen- and strontium-doped compounds with similar $T_text{c}$, we rule out a connection between the phonon anomaly and structural instabilities related to the specific dopant type. We interpret the phonon anomaly as a signature of correlated charge fluctuations possibly connected to stripes.
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