  The mixed braid groups $B_{2,n}, \ n \in \mathbb{N}$, with two fixed strands
and $n$ moving ones, are known to be related to the knot theory of certain
families of $3$-manifolds. In this paper we define the mixed Hecke algebra
$\mathrm{H}_{2,n}(q)$ as the quotient of the group algebra ${\mathbb Z}\,
[q^{\pm 1}] \, B_{2,n}$ over the quadratic relations of the classical
Iwahori-Hecke algebra for the braiding generators. We furhter provide a
potential basis $\Lambda_n$ for $\mathrm{H}_{2,n}(q)$, which we prove is a
spanning set for the $\mathbb{Z}[q^{\pm 1}]$-additive structure of this
algebra. The sets $\Lambda_n,\ n \in \mathbb{Z}$ appear to be good candidates
for an inductive basis suitable for the construction of Homflypt-type
invariants for knots and links in the above $3$-manifolds.
