We present the first weak-lensing mass calibration and X-ray scaling relations of galaxy clusters and groups selected in the $eROSITA$ Final Equatorial Depth Survey (eFEDS) observed by Spectrum Roentgen Gamma/$eROSITA$ over a contiguous footprint with an area of $approx140$ deg$^2$, using the three-year (S19A) weak-lensing data from the Hyper Suprime-Cam (HSC) Subaru Strategic Program survey. In this work, a sample of $434$ optically confirmed galaxy clusters (and groups) at redshift $0.01lesssim z lesssim1.3$ with a median of $0.35$ is studied, of which $313$ systems are uniformly covered by the HSC survey to enable the extraction of the weak-lensing shear observable. In a Bayesian population modelling, we perform a blind analysis for the weak-lensing mass calibration by simultaneously modelling the observed count rate $eta$ and the shear profile $g$ of individual clusters through the count rate-to-mass-and-redshift ($eta$--$M_{500}$--$z$) and weak-lensing mass-to-mass-and-redshift ($M_{mathrm{WL}}$--$M_{500}$--$z$) relations, respectively, while accounting for the bias in these observables using simulation-based calibrations. As a result, the count rate-inferred and lensing-calibrated cluster mass is obtained from the joint modelling of the scaling relations, as the ensemble mass spanning a range of $10^{13}h^{-1}M_{odot}lesssim M_{500}lesssim10^{15} h^{-1}M_{odot}$ with a median of $approx10^{14} h^{-1}M_{odot}$ for the eFEDS sample. With the mass calibration, we further model the X-ray observable-to-mass-and-redshift relations, including the rest-frame soft-band and bolometric luminosity ($L_{mathrm{X}}$ and $L_{mathrm{b}}$), the emission-weighted temperature $T_{mathrm{X}}$, the mass of intra-cluster medium $M_{mathrm{g}}$, and the mass proxy $Y_{mathrm{X}}$, which is the product of $T_{mathrm{X}}$ and $M_{mathrm{g}}$. (abridged)
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