  We revisit the free field construction of the deformed $W$-superalgebras
${\cal W}_{q,t}(\mathfrak{sl}(2|1))$ by J. Ding and B. Feigin, {\it
Contemp.Math.}{\bf 248}, 83-108 (1998), where the basic $W$-current and
screening currents have been found. In this paper we introduce higher
$W$-currents and obtain a closed set of quadratic relations among them. These
relations are independent of the choice of Dynkin diagrams for the superalgebra
$\mathfrak{sl}(2|1)$, though the screening currents are not. This allows us to
define ${\cal W}_{q,t}(\mathfrak{sl}(2|1))$ by generators and relations.
