Is it possible to classify Hopf algebras $H$, over a field $k$, which admit a unique (up to isomorphism) irreducible comodule, namely the trivial $1$-dim comodule
$$
k \to k \otimes H, ~~ k \mapsto k \otimes 1\_H.
$$



Classifying Hopf algebras that admit a single irreducible comodule
The HAs you are describing are again the connected (=irreducible) ones. I am using the terminology here as in my answer to your previous question: [Name for a Hopf algebra whose only grouplike element is the identity?](https://mathoverflow.net/q/400726/85967).


Their classification, is in general an open project:


* In the cocommutative case, over a field of $chark=0$, they are all universal enveloping algebras of lie algebras. If $k=\mathbb{C}$, this is by an old result of Milnor and Moore. You can find the statement for any field of char zero at Montgomery's book.
* In the finite dimensional case they appear only over fields of positive characteristic. I think this is an old result of Masuoka (but i do not have the exact reference right now). See [arXiv:1309.0286 [math.RA]](https://arxiv.org/abs/1309.0286), [arXiv:1310.7073 [math.RA]](https://arxiv.org/abs/1310.7073) (and the references therein) for recent results on their classification.


