Knot Floer homology of satellite knots with (1,1)-patterns

For pattern knots admitting genus-one bordered Heegaard diagrams, we show the knot Floer chain complexes of the corresponding satellite knots can be computed using immersed curves. This, in particular, gives a convenient way to compute the $\tau$-invariant. For patterns $P$ obtained from two-bridge links $b(p,q)$, we derive a formula for the $\tau$-invariant of $P(T_{2,3})$ and $P(-T_{2,3})$ in terms of $(p,q)$, and use this formula to study whether such patterns induce homomorphisms on the concordance group, providing a glimpse at a conjecture due to Hedden.