It's a relative clause. Here's the non-relative version:
He says the conviction was politically motivated.
And the relative versions:
1a. a conviction [ which he says ____ was politically motivated ]
1b. a conviction [ that he says ____ was politically motivated ]
1c. a conviction [ he says ____ was politically motivated ]
All three are possible, thanks to some technical rules:
- It's an restrictive relative, so we can replace which with that.
- The gap is not in subject position, so we can omit that.
- This is not a stacked relative. In stacked relatives, that can only be omitted from the first.
This particular example is a little complicated because the relative clause itself contains a content clause, which could normally be marked with that:
He says [ that the corruption conviction was politically motivated ].
But in the relative versions, inserting that after says is not possible:
2a. a conviction [ which he says [
that ____ was politically motivated ] ]
2b. a conviction [ that he says [ that ____ was politically motivated ] ]
2c. a conviction [ he says [ that ____ was politically motivated ] ]
Strange, isn't it? Why do you suppose that is?
The Cambridge Grammar of the English Language (CGEL) uses a special name for this sort of thing. They call it an unbounded dependency construction. The basic idea behind this term is that the dependency between the gap and the noun outside the relative clause can go through as many layers of embedding as you want.
In other words, it's unbounded. The gap doesn't have to appear directly in the relative clause itself. It can appear inside an embedded clause instead!
In this case, it appears inside an embedded content clause in subject position. And when that's the case, that must be omitted. Take a look at these examples from CGEL, p.1083:
3a. He's the man [ they think [
that ____ attacked her ] ].
3b. *He's the man [ they think [ that ____ attacked her ] ].
Here, the second example is ungrammatical because of that. In this construction, it must be omitted!
So as you can see, your sentence is indeed a bit tricky grammatically. But once you learn to recognize the structure, it should get easier to understand.