  The polarized reflectivity of
$\beta^{\prime\prime}$-(BEDO-TTF)$_5$[CsHg(SCN)$_4$]$_2$ is studied in the
infrared range between 60 cm-1 and 6000 cm-1 from room temperature down to 10
K. Already at T=300 K a pseudogap in the optical conductivity is present of
about 300 cm-1; the corresponding maximum in the spectrum shifts to lower
frequencies as the temperature decreases. In contrast to quarter-filled
BEDT-TTF-based conductors of the $\beta^{\prime\prime}$-phase a robust Drude
component in the conductivity spectra is observed which we ascribe to the
larger fraction of charge carriers associated with the 1/5-filling of the
conduction band. This observation is corroborated by exact diagonalization
calculations on an extended Hubbard model on a square lattice for different
fillings. A broad band at 4000 cm-1 which appears for the electric field
polarized parallel to the stacks of the BEDO-TTF molecules is associated to
structural modulations in the stacks; these modulations lead to a rise of the
dc and microwave resistivity in 100 to 30 K temperature range.
