David is right: there isn't yet a general theory of "mixed automorphic representations". Let me propose some ideas of what a theory could be.

First, I will work with the $p$-adic realizations of mixed motives, rather than with the the mixed motives themself, for concreteness and because motives are harder to construct that Galois representations.

I don't think that it is reasonable to expect that every $p$-adic non-semisimple Galois representations (with Fontaine-Mazur-like conditions) should come from an automorphic representations. In fact, I don't even know what "come" would mean here : to an algebraic automorphic representation, say for $Gl_n$ or for a reductive group G together with a representation of its $L$-group into $Gl_n$, I know at least conjecturally how to attach a **semi-simple** Galois representations, not a non-trivial extension.

Rather, I expect non semi-simple Galois representations (with suitable local conditions that makes them "motivic") would all come from **$p$-adic deformations
of automorphic form** (in the sense of eigenvarieties). Rather than to try to formulate this in general (that would be mostly guess-work), let me consider an interesting very particular case.

Let $F$ be a number field. I consider extensions of $\mathbb Q_p$ by $\mathbb Q_p(1)$ in the category of $G_F$-representations, which are unramified outside $p$ and crystalline at $p$. The spaces of such extensions is naturally isomorphic, as is well-known (Kümmer Theory ++), to the space $\cal O_F^\ast \otimes_{\mathbb Z}
\mathbb Q_p$, whose dimension is determined by Dirichlet's unit Theorem to be $[F:\,mathbb Q]-1$.

Say $F$ is totally real to fix ideas (and different from $\mathbb Q$) because in this case the eigenvariety of over convergent $p$-adic automorphic forms for $GL_2/F$ has been constructed (Andreatta-Iovita-Pilloni-Stevens).
There is a (unique) point $x$ on this eigenvariety corresponding on this the Eisenstein series $E_2$ over $F$ with critical refinements at every place above $p$.

The eigenvariety carries a family of Galois representations (rather a pseudo-representation) which at $x$ is the (semi-simple) representation $\mathbb Q_p \oplus \mathbb Q_p(1)$ but is irreducible at points near $x$ (that's a consequence of the choice of "critical" refinements). If you choose any curve $C$ in the eigenvariety through $x$, smooth at $x$, and restrict your family of Galois representation to $C$, and even to the local ring of $C$ at $x$, then the famous "Ribet lemma" gives you an extension of $\mathbb Q_p$ by $\mathbb Q_p(1)$ in the category of Galois representations of $G_F$, unramified outside $p$ and crystalline at $p$.

We can say that this extension is (the $p$-adic realization) of a mixed motive, automorphically constructed.

Then in this special case, you question can be precisely formulated:

**Conjecture**: All such extensions of $\mathbb Q_p$ by $\mathbb Q_p(1)$ comes from a deformation of $E_2$ this way.

I don't know if that's true beyond the trivial $[F:\mathbb Q]=2$ case. But I think it is reasonable to expect this. A student of me, Yu Fang, proved it is true when the eigenvariety is smooth at $x$.