We give a modification of Yu’s construction of supercuspidal representations of a connected reductive group G over a non-Archimedean local field F. This modification restores the validity of certain key intertwining property claims made by Yu, which were recently proved to be false for the original construction. This modification is also an essential ingredient in the construction of supercuspidal L-packets in a preprint by the second author. As further applications, we prove the stability and many instances of endoscopic character identities of these supercuspidal L-packets, subject to some conditions on the base field F. In particular, for regular supercuspidal parameters, we prove all instances of standard endoscopy. In addition, we prove that these supercuspidal L-packets satisfy a recent conjecture by the second author, which, together with standard endoscopy, uniquely characterizes the local Langlands correspondence for supercuspidal L-packets (again subject to the conditions on F). These results are based on a statement of the Harish-Chandra character formula for the supercuspidal representations arising from the twisted Yu construction.