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Hurwitz numbers with completed cycles are standard Hurwitz numbers withsimple branch points replaced by completed cycles. In fact, simple branchpoints correspond to completed $2$-cycles. Okounkov and Pandharipande haveestablished the remarkable GW/H correspondence, saying that the stationarysectors of the Gromov--Witten theory relative to $r$ points equal Hurwitznumbers with $r$ branch points besides the completed cycles. However, from theviewpoint of computation, known results for Hurwitz numbers (standard or withcompleted cycles) are mainly for $r\leq 2$. It is hard to obtain explicitformulas and then discuss the structural properties for the cases $r>2$. Inthis paper, we obtain explicit formulas for the case $r=3$ and uncover a numberof structural properties of these Hurwitz numbers. For instance, we discover apiecewise polynomiality with respect to the orders of the completed cycles inaddition to the parts of the profiles of branch points as usual, we show thatcertain hook-shape Hurwitz numbers are building blocks of all our Hurwitznumbers, and we prove an analogue of the celebrated $\lambda_g$-conjecture.

Hurwitz numbers with completed cycles and Gromov--Witten theory relative to at most three points