  In this article, a finite analogue of the generalized sum-of-tails identity
of Andrews and Freitas is obtained. We derive several interesting results as
special cases of this analogue, in particular, a recent identity of Dixit,
Eyyyunni, Maji and Sood. We derive a new extension of Abel's lemma with the
help of which we obtain a one-parameter generalization of a sum-of-tails
identity of Andrews, Garvan and Liang, an identity of Ramanujan as well as two
new results - one for Ramanujan's function $\sigma(q)$ and another for the
function recently introduced by Andrews and Ballantine. Later we introduce a
new generalization $\mathrm{FFW}_{c}(n)$ of a function of Fokkink, Fokkink and
Wang and derive an identity for its generating function. This gives, as a
special case, a recent representation for the generating function of
$\mathrm{spt}(n)$ given by Andrews, Garvan and Liang. We also obtain some
weighted partition identities along with new representations for two of
Ramanujan's third order mock theta functions through combinatorial techniques.
