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We present the transit timing variation (TTV) and planetary atmosphere analysis of the Neptune-mass planet HAT-P-26~b. We present a new set of 13 transit light curves from optical ground-based observations and combine them with light curves from the Wide Field Camera 3 (WFC3) on the Hubble Space Telescope (HST), Transiting Exoplanet Survey Satellite (TESS), and previously published ground-based data. We refine the planetary parameters of HAT-P-26 b and undertake a TTV analysis using 33 transits obtained over seven years. The TTV analysis shows an amplitude signal of 1.98 $\pm$ 0.05 minutes, which could result from the presence of an additional $0.02 M_{Jup}$ planet at the 1:2 mean-motion resonance orbit. Using a combination of transit depths spanning optical to near-infrared wavelengths, we find that the atmosphere of HAT-P-26 b contains $2.4^{+2.9}_{-1.6}$% of H$_2$O with a derived temperature of $590^{+60}_{-50}$ K.