You are probably thinking about undecidable languages which are computably enumerable. Otherwise, the diagonalization technique described in answers to similar questions would have provided simple counterexamples. If you don't care about the computably enumerable part, then I would say that your question is simply a duplicate of one of the similar questions.
The halting problem is at least as strong than any such language, because you can just fix the string for which you want to know whether it is in the language, define a Turing machine which enumerates the strings in the language until it finds the given string and then stops, and then ask the halting oracle whether that Turing machine will stop.
So are there such languages which are weaker than the halting problem and still undecidable? Yes, there are (at least ZFC claims that there are, and most other reasonable formal systems will think so too). Can I write down such a language? No, not at the moment. Does anybody knows how to write down such a language? Maybe, I don't know.
Why do I write an answer, if I can't really provide an explicit answer? Because I want to point out a connection to formal systems. A language strong enough to complement a given axiom system (able to talk about TM like Peano arithmetic (PA) for example) to a consistent set of axioms complete for $\Pi^0_1$ sentences (i.e. the halting problem) provides enough computational power to construct a model of the given axiom system, and the set of $\Pi^0_1$ sentences of any model gives such language. The following answer by Noah Schweber first defines what it means to complement PA (as an example)
A set $A$ of (indices for) $\Pi^0_1$ sentences is plausible if $PA\cup A\cup\{\neg \varphi: \varphi\in\Pi^0_1\setminus A\}$ is consistent.
And then proceeds to show how Rosser's trick can be used to show
I claim that every plausible set $A$ is of PA degree
i.e. construct a model of the given axiom system (PA in this case). And if you had a stronger axiom system like ZFC, then being plausible with respect to ZFC would define you a ZFC degree, which is at least as strong as the PA degree. (I guess there are languages of PA degree which are not of ZFC degree, but in this case I don't even know whether ZFC claims this.)