Dear Mohamed,

the nature of the pseudogap in brief: T* is the pairing temperatur (or, equivalently, temperature of spin singlet formation). At T between Tc and T*, the spin singlets are present, but they are not condensed into Bose-Einstein-condensate (BEC), because Tbec (temperature of the BEC) is lower than than T*. This regime is possible, for example, in underdoped cuprates, since there the spin correlation is strong and the pair density is small (Tbec is directly related to the pair density). In Fermi metals always Tbec=T*, so there is no pseudogap. This case is shown in Eq. (3.13) in https://arxiv.org/ftp/arxiv/papers/1501/1501.04978.pdf

Thus the pseudogap, as pairing strength, is related with the superconductivity. However, if the pairing strength is very strong, Tc may be low because of low electron density. Usually, the stronger the singlet correlation, the lower the electron density (and, of course, the pair density is also low).

A pairing gap in the quasi-particle excitation spectrum is a necessity to understand the occurrence of superconductivity. The reason for high temperature superconductivity is not well understood so far. However, the Coulomb correlation is believed to be a very important reason in several published papers. The exact process through which the Coulomb correlation affects the Tc is yet to be discovered.

A pseudogap (gap in energy) is not associated with true superconducting state (defined by zero electrical resistance and perfect diamagnetism) in HTSC unlike a pairing gap in most other superconductor we know.