We suggest a new hardcore Poisson-type distribution for Young diagrams with the row lengths from some finite list. A discrete variant of the time-ordered Matérn II process in 1D is employed. This approach is related to that based on the interlacing sequences due to Kerov and others, but we restrict the number of rows. The basic lengths are assumed comparable with the total order of the diagram in the quasi-classical limit, which results in new methods and new formulas. An interesting application is to random walks where the steps are at the points satisfying the classical Poisson distribution or our truncatedone. In the simplest case, one obtains the distribution in terms of Bessel I-functions, which provides some probabilistic interpretation of its many properties. An immediate application of our truncated Poisson distributions is to modeling reinfections in epidemics.