  The wavelet-based detrending and denoising method \texttt{TFAW} is applied
for the first time to \texttt{EVEREST 2.0}-corrected light curves to further
improve the photometric precision of almost all K2 observing campaigns (C1-C8,
C12-C18). The performance of both methods is evaluated in terms of 6 hr
combined differential photometric precision (CDPP), simulated transit detection
efficiency, and planet characterization in different SNR regimes. On average,
\texttt{TFAW} median 6hr CDPP is $\sim$30$\%$ better than the one achieved by
\texttt{EVEREST 2.0} for all observing campaigns. Using the \texttt{transit
least-squares} (\texttt{TLS}) algorithm, we show that the transit detection
efficiency for simulated Earth-Sun-like systems is $\sim$8.5$\times$ higher for
\texttt{TFAW}-corrected light curves than for \texttt{EVEREST 2.0} ones. Using
the light curves of two confirmed exoplanets, K2-44 b (high-SNR) and K2-298 b
(low-SNR), we show that \texttt{TFAW} yields better MCMC posterior
distributions, transit parameters compatible with the cataloged ones but with
smaller uncertainties and narrows the credibility intervals. We use the
combination of \texttt{TFAW}'s improved photometric precision and \texttt{TLS}
enhancement of the signal detection efficiency for weak signals to search for
new transit candidates in K2 observing campaign 1. We report the discovery of
two new K2-C1 Earth-sized planets statistically validated, using the
\texttt{vespa} software: EPIC 201170410.02, with a radius of
1.047$^{+0.276}_{-0.257}R_{\oplus}$ planet orbiting an M-type star, and EPIC
201757695.02, with a radius of 0.908$^{+0.059}_{-0.064}R_{\oplus}$ planet
orbiting a K-type star. EPIC 201757695.02 is the 9-th smallest planet ever
discovered in K2-C1, and the 39-th smallest in all K2 campaigns.
