We study the manipulation of Goos-H\"anchen (GH) shifts for the reflected and transmitted probe light pulses injected into a cavity containing four-level configuration mercury atoms where the probe transition is in the ultraviolet (UV) region with a wavelength of
253.7nm. Different behaviors of the GH shifts can be observed in the absence, or presence, of two driving fields as well as an incoherent pump field. When neither coherent driving fields nor incoherent pumping is turned on, we realize negative reflected GH shifts for anomalous dispersion. Including only one driving field leads to subluminal-based light propagation with positive lateral shifts at certain incident angles. Taking into account the impact of both driving fields, negative GH shifts reappear in the reflected part of the incident light. The origin of this defect is attributed to interacting double dark resonances due to a high-resolution absorption peaks with a very steep negative slope of dispersion in the susceptibility profile. We then show that one can surpass this defect by applying a weak incoherent pumping field to obtain positive GH shifts for both reflected and transmitted light beams. Finally, using the $6{\phantom{\rule{0.16em}{0ex}}}^{1}{P}_{1}\ensuremath{\leftrightarrow}6{\phantom{\rule{0.16em}{0ex}}}^{1}{S}_{0}$ transition of Hg, we generalize our study to the case where the wavelength of the probe transition is
185nm which is in the vacuum-ultraviolet domain. Although the number of oscillations is now increased, however, similar results are reported with respect to the case of UV transition.
The role of a chiral quantum-dot molecule (QDM) system in controlling and enhancing the Goos-H\"anchen (GH) shift is revealed here. The proposed setup is a four-level atomic system excited by incoherent pump and probe fields where the electron tunneling is prominent in affecting the GH shift upon the reflection and transmission of probe beams. The GH shifts in the chiral QDM system are calculated analytically by using stationary phase theory, whereas the dynamics of the quantum-dot system are derived by the density-matrix method. It is demonstrated that the GH shifts through transmission and reflection light beams are strongly influenced due to right-circularly polarized (RCP) and left-circularly polarized light. The GH shifts through RCP light are enhanced in the presence of chiral QDM systems, and these shifts could become either highly negative or highly positive, sensitively depending on the electron tunneling and chiral coefficients. This proposal can be viewed as a theoretical approach to the potential application of GH shifts in nano-optoelectronic sensors.