This is a hard question to answer for a number of reasons. One of which is to say: how would we know? The total sample size of historical cases is very small (only a handful of countries have nuclear weapons, and of those, only a few for sure have thermonuclear weapons), and inferring diplomatic "power" from such a small and specific sample size is difficult. Which is to say, one might want to make an argument that, say, North Korea and Pakistan have different negotiating power with the US — what accounts for that difference? Obviously a lot of differences exist between the two countries and their relationships to the US. How to isolate the relative power of them having different types of nuclear weapons? It's not clear to me that you could isolate such things (though that has hardly stopped quantitative International Relations folks from trying to do similar sorts of studies). It is very hard to disentangle the technical difference of thermonuclear and fission bombs from other technical advances that usually happened simultaneously. In the case of North Korea, for example, the biggest apparent difference is not whether they had fission or thermonuclear warheads (that matters a little), but whether they had ICBMs (that matters a lot). The difference between a 15 kiloton weapon and a 150 kiloton weapon, in terms of damage, is not as great as the difference between having a reliable ICBM and not having one. In the former case, the difference is about twice the area affected — not nothing, to be sure, but not the 10X difference that it first appears — and in the latter case, it's the difference between them being able to threaten the US mainland and them not being able to.

From a historical-strategic standpoint, what we can say is this: what hydrogen bombs really give a nation is not just an increased destructive power, which may or may not matter from the perspective of the other side, but it makes certain delivery vehicles more likely to hit their targets. Thermonuclear weapon design allows you to put a relatively large amount of force into a relatively small package, which means that you can compensate for the inaccuracies of ballistic missiles. So if you only have a fission weapon on a ballistic missile, it has to be very accurate. If you have a thermonuclear weapon, it does not. Thus, in the language of the nuclear game theorists, it becomes a more "credible" threat.

Does that actually matter from a negotiating standpoint? That is not clear. Do heads of state think in game theory terms? It's clear that they often don't — US and Soviet heads of state repeatedly emphasized behind closed doors (we learned after the fact) that they didn't trust this kind of thinking and instead basically conceptualized nuclear weapons as a form of total destruction. The difference between a state having a 50% chance of destroying a given target and a 70% chance was not seen by these people as meaningful, even though it was sometimes seen as meaningful by strategic planners and theorists (often in ways that suited their ideological interests).

However, there are cases where we have evidence that at least some national heads of state saw the difference as being one of technological prestige. So France pushed for thermonuclear weapons not so much because they were essential to its strategic position, but because they thought they'd be treated as a "lesser" state if China (a poorer and less technologically-advanced state at the time) had them and they did not.

Similarly it is clear that many US policymakers believed that if the USSR had the H-bomb and they did not, then the USSR would have huge leverage over the USA. There were many military and scientific advisors at the time who disagreed with this view, given the fact that the US had about a 100 to 1 advantage in fission weapons over the USSR, and (unlike the USSR) actually had the means to deliver them to the enemy's cities and headquarters, and thus saw the difference in yield as relatively unimportant. But in weighing political power, it matters who one is talking about: the opinions of scientists tend to matter less than Congressmen and Presidents when it comes to international negotiations. And when it comes to very high-level negotiations (e.g. President to President), then individual psychology matters even more — and in such cases, really anything can be on the table.

Anyway, this is an interesting question. I am not sure if it is answerable from either a historical or political science perspective, but I've tried to sketch out some of the aspects of it.