  The eROSITA Final Equatorial-Depth Survey (eFEDS), executed during the
performance verification phase of the Spectrum-Roentgen-Gamma (SRG)/eROSITA
telescope, was completed in Nov. 2019. One of the science goals of this survey
is to demonstrate the ability of eROSITA to detect samples of clusters and
groups at the final depth of the eROSITA all-sky survey. Because of the
sizeable point-spread function of eROSITA, high-redshift clusters of galaxies
or compact nearby groups hosting bright active galactic nuclei (AGN) can be
misclassified as point sources by the source detection algorithms. A total of
346 galaxy clusters and groups in the redshift range of 0.1<z<1.3 were
identified based on their red sequence in the point source catalog. We examine
the multiwavelength properties of these clusters and groups to understand the
potential biases in our selection process and the completeness of the
extent-selected sample. The majority of the clusters and groups in the point
source sample are indeed underluminous and compact compared to the
extent-selected sample. Their faint X-ray emission, well below the flux limit
of the extent-selected eFEDS clusters, and their compact X-ray emission are
likely to be the main reason for this misclassification. In the sample, we
confirm that 10% of the sources host AGN in their brightest cluster galaxies
(BCGs) through optical spectroscopy and visual inspection. By studying their
X-ray, optical, infrared, and radio properties, we establish a method for
identifying clusters and groups that host AGN in their BCGs. We successfully
test this method on the current point source catalog through the Sloan Digital
Sky Survey optical spectroscopy and find eight clusters and groups with active
radio-loud AGN that are particularly bright in the infrared. They include
eFEDSJ091437.8+024558, eFEDSJ083520.1+012516, and eFEDSJ092227.1+043339 at
redshifts 0.3-0.4. [ABRIDGED]
