  I attempt to analyse the next-to-leading-order non-holomorphic contribution
to the Wilsonian low-energy effective action in the four-dimensional N=2 gauge
theories with matter, from the manifestly N=2 supersymmeric point of view, by
using the harmonic superspace. The perturbative one-loop correction is found to
be in agreement with the N=1 superfield calculations of de Wit, Grisaru and
Rocek. The previously unknown coefficient in front of this non-holomorphic
correction is calculated. A special attention is devoted to the N=2
superconformal gauge theories, whose one-loop non-holomorphic contribution is
likely to be exact, even non-perturbatively. This leading (one-loop)
non-holomorphic contribution to the LEEA of the N=2 superconformally invariant
gauge field theories is calculated, and it does not vanish, similarly to the
case of the N=4 super-Yang-Mills theory.
