  An immersed concordance between two links is a concordance with possible
self-intersections. Given an immersed concordance we construct a smooth
four-dimensional cobordism between surgeries on links. By applying
$d$-invariant inequalities for this cobordism we obtain inequalities between
the $H$-functions of links, which can be extracted from the link Floer homology
package. As an application we show a Heegaard Floer theoretical criterion for
bounding the splitting number of links. The criterion is especially effective
for L-space links, and we present an infinite family of L-space links with
vanishing linking numbers and arbitrary large splitting numbers. We also show a
semicontinuity of the $H$-function under $\delta$-constant deformations of
singularities with many branches.
