First of all, as pointed out in the other answer, The gamma matrices satisfy $\gamma^\mu\gamma_\mu=4\mathbb{I}$, so the $\lambda$ term reduces to the mass term.
I am concentrating on the $\alpha$ term:
$$
\begin{equation}
\bar{\Psi}_{k}(\alpha D^{\mu}D^{\nu}\Omega^{kj}_{\mu\nu})\Psi_{j}
\end{equation}
$$
Contrary to the claim of the other answer, such term with two derivatives is allowed, but there is a catch!
1, Symmetric case of $\Omega^{kj}_{\mu\nu}$
Let's look at the specific case of $\Omega^{kj}_{\mu\nu} = \eta_{\mu\nu}\delta^{kj}$ and ignore gauge field coupling, then the term reduces to the Klein–Gordon-like (applying simplified notation than OP)
$$
\begin{equation}
\alpha \bar{\psi}\partial^\mu\partial_\mu\psi
\end{equation}
$$
The modified/enhanced Dirac Lagrangian can be written as
$$
L = i\bar{\psi}\not D\psi + \alpha \bar{\psi}\partial^\mu\partial_\mu\psi- m\bar{\psi}\psi \tag{1}
$$
where the Klein–Gordon-like term $\alpha\bar{\psi}\partial^\mu\partial_\mu\psi$ is added to the original Dirac-like term $i\bar{\psi}\not D\psi$.
The Klein–Gordon-like term $\bar{\psi}\partial^\mu\partial_\mu\psi$ is a dimension-5 operator (3 from the the two dimension-$3/2$ spinors plus 2 from the two derivatives), and thus non-renormalizable. In principle, such non-renormalizable term IS allowed in the Effective Field Theory framework. However, given that the Klein–Gordon-like term is a dimension-5 operator, it is suppressed by a factor of:
$$
\frac{E}{M_{Planck}}
$$
where $E$ is the energy scale of the physics process in concern, and $ M_{Planck}$ is the Planck scale.
In other words, the Klein–Gordon-like term can be safely ignored under normal circumstances (i.e. when $E << M_{Planck}$), unless we are dealing with Planck energy scale physics such as early universe during big bang.
2, Anti-Symmetric case of $\Omega^{kj}_{\mu\nu}$
As for the other tensors form of $\Omega^{kj}_{\mu\nu}$, the anti-symmetrical portion of the gauge-covariant derivative $D^{\mu}D^{\nu}$ reduces to the 'non-minimal' interaction term originally suggested by Pauli:
$$
\bar{\psi}F^{\mu\nu}\sigma_{\mu\nu}\psi
$$
where $F^{\mu\nu}$ is the electromagnetic field tensor, and $\sigma_{\mu\nu}$ are Pauli matrices.
The Pauli term is dimention-5 too, therefore faces the same Planck-scale suppression issue as we discussed for the Klein–Gordon-like term.
3, Technical naturalness issue
Additionally, both the Pauli term $\bar{\psi}F^{\mu\nu}\sigma_{\mu\nu}\psi$ and the Klein–Gordon-like term $\bar{\psi}\partial^\mu\partial_\mu\psi$ break the axial symmetry
$$
\psi \rightarrow e^{\theta i\gamma_5}\psi
$$
Hence these terms are further suppressed due to t' Hooft's technical naturalness argument.
4, Other cases not considered in OP
And for that matter, there are other terms not considered in OP, such as three derivative term
$$
\bar{\psi}D^{\mu}D^{\nu}\gamma_\mu D_{\nu}\psi \sim \bar{\psi}F^{\mu\nu}\gamma_{\mu}D_{\nu}\psi
$$
or four derivative term
$$
\bar{\psi}D^{\mu}D^{\nu}D_\mu D_{\nu}\psi \sim \bar{\psi}F^{\mu\nu}F_{\mu\nu}\psi
$$
I will leave it to the readers as a home work excise as to whether they are (not) allowed.
See more details here.
