This is the problem $(\beta)$ of the section 14.7 in the "Analytical Guide" of the Shelah's book "Cardinal Arithmetic":

$(\beta)$ Is $\operatorname{cov} ( \lambda , \lambda, \aleph_{1} , 2) =^{+} \operatorname{pp} ( \lambda )$ when $\operatorname{cf}( \lambda ) = \aleph_0$?

Is this problem still open?

Can we have $\operatorname{cov} ( \lambda , \lambda, {(\operatorname{cf}(\lambda))}^{+} , 2) = \operatorname{pp} ( \lambda )$ for every singular $\lambda$?

We know that $\operatorname{cov} ( \lambda , \lambda, {(\operatorname{cf}(\lambda))}^{+} , 2) = \operatorname{pp} (\lambda )$ for every singular $\lambda$ such that $\lambda < \aleph_\lambda$ (see for instance the Claim 3.7(1) in the Chapter IX of the Shelah's book).