  The Sunyaev-Zeldovich (SZ) effect provides a powerful cosmological probe,
which traditionally is approached independently as cluster number count (CNC)
or power spectrum (PS) analysis. Here, we devise a new method for analysing the
$y$-map by introducing the survey completeness function, conventionally only
used in the CNC analysis, in the $yy$-PS modeling. This provides a systematic
method, based mainly on SZ observables, for obtaining two complementary
$y$-maps, one incorporating detected/resolved clusters and the other relying
only on diffuse/unresolved SZ contributions. We use the catalogue of clusters
obtained in the \Planck CNC analysis to define the completeness function
linking these two $y$-maps. The split depends on the chosen signal-to-noise
detection threshold, which we vary in our discussion. We carefully propagate
the effect of completeness cuts on the non-Gaussian error contributions in the
$yy$-PS analysis, highlighting the benefits of masking massive clusters. Our
analysis of the \Planck $yy$-PS for the unresolved component yields a mass bias
of $b=0.15\pm0.04$, consistent with the standard value ($b\approx0.2$), in
comparison to $b=0.4\pm 0.05$ for the total $yy$-PS. We find indications for
this drift being driven by the CIB-tSZ cross correlation, which dominantly
originates from clusters in the resolved component of the $y$-map. Another
possible explanation is the presence of a mass-dependent bias, which has been
theoretically motivated and can be quantified with our novel method. We
furthermore find first hints for the presence of the 2-halo terms in the
$yy$-PS. Finally, the proposed method provides a new framework for combining
the complementary information of the CNC and PS analyses in upcoming SZ
surveys.
