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Let 1 ≤ p ≤ ∞ and let $w_0,w_1$ be two weights on the unit circle such that $log\left(w_0{w}_1^{-1}\right)\in BMO$. We prove that the couple $(H_p\left(w_0\right),H_p\left(w_1\right))$ of weighted Hardy spaces is a partial retract of $(L_p\left(w_0\right),L_p\left(w_1\right))$. This completes previous work of the authors. More generally, we have a similar result for finite families of weighted Hardy spaces. We include some applications to interpolation.