  We study the affine analogue $\mathrm{FT}_p(\mathfrak{sl}_2)$ of the triplet
algebra. We show that $\mathrm{FT}_p(\mathfrak{sl}_2)$ is quasi-lisse and the
associated variety is the nilpotent cone of $\mathfrak{sl}_2$. We realize
$\mathrm{FT}_p(\mathfrak{sl}_2)$ as the global sections of a sheaf of vertex
algebras in the spirit of Feigin--Tipunin and thereby construct infinitely many
simple modules and, in particular solve a conjecture by Semikhatov and Tipunin.
We introduce the Kazama--Suzuki dual superalgebra
$s\mathcal{W}_p(\mathfrak{sl}_{2|1})$ of $\mathrm{FT}_p(\mathfrak{sl}_2)$ and
their singlet type subalgebras $s\mathcal{M}_p(\mathfrak{sl}_{2|1})$ and
$\mathcal{M}_p(\mathfrak{sl}_2)$ and show their correspondence of categories.
For $p=1$, we show the logarithmic Kazhdan--Lusztig correspondence for these
(super)algebras and, in particular, show that the quantum group corresponding
to $s\mathcal{M}_1(\mathfrak{sl}_{2|1})$ is the unrolled restricted quantum
supergroup $u^H_{-1}(\mathfrak{sl}_{2|1})$ as suggested by Semikhatov and
Tipunin.
