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Throughout the passage within the Galactic halo, high-velocity clouds (HVCs) sweep up ambient magnetic fields and form stretched and draped configurations of magnetic fields around them. Many earlier numerical studies adopt spherically symmetric uniform-density clouds as initial conditions for simplicity. However, observations demonstrate that HVCs are clumpy and turbulent. In this paper, we perform 3D magnetohydrodynamic simulations to study the evolution of clouds with initial density distributions described by power-law spatial power spectra. We systematically study the role of (i) the initial density structure, (ii) halo magnetic fields, and (iii) radiative cooling efficiency upon infalling HVCs. We find that (i) the clouds' density structure regulates mixing and mass growth. Uniform clouds grow from the onset of the simulations while clumpy clouds initially lose gas and then grow at later times. Along the same lines, the growth curve of clumpy clouds depends on the slope of the initial density power spectra. (ii) Magnetic fields suppress hydrodynamic instabilities and the growth of small-scale structures. As a result, magnetized clouds develop long filaments extended along the streaming direction whereas non-magnetized clouds are fragmented into many small clumps. (iii) Efficient cooling keeps the main cloud body more compact and produces decelerated dense clumps condensed from the halo gas. This work potentially helps us understand and predict the observed properties of HVCs such as the detectability of magnetized clouds, the presence of decelerated HI structures associated with HVC complexes and small-scale features, and a possible link between the origin and the fate of HVCs.