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

$(\beta)$ Is $cov ( \lambda , \lambda, \aleph_{1} , 2) =^{+} pp ( \lambda )$ when $cf ( \lambda ) = \aleph_{0}$?

Is this problem still open?

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

We know that $cov ( \lambda , \lambda, {(cf(\lambda))}^{+} , 2) = pp ( \lambda )$ 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).