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Searching for Earth-sized planets in data from Kepler's extended mission (K2) is a niche that still remains to be fully exploited. The TFAW survey is an ongoing project that aims to re-analyze all light curves in K2 C1-C8 and C12-C18 campaigns with a wavelet-based detrending and denoising method, and the period search algorithm TLS to search for new transit candidates not detected in previous works. We have analyzed a first subset of 24 candidate planetary systems around relatively faint host stars (10.9 < $K_{p}$ < 15.4) to allow for follow-up speckle imaging observations. Using VESPA and TRICERATOPS, we statistically validate six candidates orbiting four unique host stars by obtaining false-positive probabilities smaller than 1% with both methods. We also present 13 vetted planet candidates that might benefit from other, more precise follow-up observations. All of these planets are sub-Neptune-sized, with two validated planets and three candidates with sub-Earth sizes, and have orbital periods between 0.81 and 23.98 days. Some interesting systems include two ultra-short-period planets, three multi-planetary systems, three sub-Neptunes that appear to be within the small planet Radius Gap, and two validated and one candidate sub-Earths (EPIC 210706310, EPIC 210768568, and EPIC 246078343) orbiting metal-poor stars., Submitted to Monthly Notices of the Royal Astronomical Society. 25 pages, 14 figures