Recent experiments indicate that several viruses may encode microRNAs (miRNAs) in cells. Such RNAs may interfere with the host mRNAs and proteins. We present a kinetic analysis of this interplay. In our treatment, the viral miRNA is considered to be able to associate with the host mRNA with subsequent degradation. This process may result in a decline of the mRNA population and also in a decline of the population of the protein encoded by this mRNA. With these ingredients, we first show the types of the corresponding steady-state kinetics in the cases of positive and negative regulation of the miRNA synthesis by the protein. In addition, we scrutinize the situation when the protein regulates the virion replication or, in other words, provides a feedback for the replication. For the negative feedback, the replication rate is found to increase with increasing the intracellular virion population. For the positive feedback, the replication rate first increases and then drops. These features may determine the stability of steady states.