  Recently, Lin introduced two new partition functions PD$_t(n)$ and
PDO$_t(n)$, which count the total number of tagged parts over all partitions of
$n$ with designated summands and the total number of tagged parts over all
partitions of $n$ with designated summands in which all parts are odd. Lin also
proved some congruences modulo 3 and 9 for PD$_t(n)$ and PDO$_t(n)$, and
conjectured some congruences modulo 8. Very recently, Adansie, Chern, and Xia
found two new infinite families of congruences modulo 9 for PD$_t(n)$. In this
paper, we prove the congruences modulo 8 conjectured by Lin and also find many
new congruences and infinite families of congruences modulo some small powers
of 2.
