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-analyse 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 analysed 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 per cent 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 23.98d. Some interesting systems include two ultra-short-period planets, three multiplanetary systems, three sub-Neptunes that appear to be within the small planet Radius Gap, and two validated and one candidate sub-Earths (EPIC 210706310.01, K2-411 b, and K2-413 b) orbiting metal-poor stars.