Do you not assume the PC is already giving it their all? If you do, how do you envision taking a deep breath helping them jump further?
PCs (in my view) aren't always giving it their all. If they were, they'd be unleashing their most empowered spells, action surging, etc every time they do something.
The way I look at this particular case is as follows:
What is the
point of the rule, under STR checks, that says a STR (Athletics) check may be made to jump further than usual? Is it to impose a strict rationing of longer jumps, or to suggest a pathway to the enabling of longer jumps? I would assume the latter, because (in my experience) longer jumps are almost never going to break the game, and enabling them - especially for players who spend PC build resources to have mighty-thewed characters - makes for fun play.
What is the
point of the rule/guideline, under "how to play", of saying that the player describes actions and then the GM determines the mode of adjudication, which may incude calling for a check? And related to that, how does a GM determine if something is uncertain? It seems to me that the point of that is to make the fiction preeminent. And a player who describes his PC as taking a deep breath and then giving it his/her all is engaging with the fiction and creating a shared experience at the table.
As far as determining whether something is uncertain, I don't know what the 5e designers had in mind and don't have much sense of how the typical 5e GM does this. But to me, saying that the outcome of the attempt to jump further than normal is not uncertain because the rules for jumping without need of a check say you can't do it seems like an overly narrow call. It seems to me that the jump rules make it
certain that you can jump X feet, thus leaving it
uncertain whether you can jump further than that - and the check is used to determine whether the further jump is possible (subject to some caveat that certain attempts at superhuman distances might just fail - presumably at most 5e tables an attempt to jump 50 feet without magical assistance is going to be declared an auto-failure by the GM).
And to combine some of the above thoughts: to require a description of what is being done to jump further than normal to be a description of different
mechanical means - like springboards or ramps - seems needlessly complex and at odds with the general spirit of the jumping rules, because if I do use a ramp to get a height advantage then why don't I get an
automatic increase in the distance I can jump (or to put it another way, why does being trained in Athletics make it easier for me to take advantage of a height difference when jumping?). And to me it also seems to encourage the fiction to focus on external elements of the characters, like what sort of equipment they can find and exploit, rather than internal elements, like how committed they are and what they are willing to risk to achieve their goals.
Frankly, I think if a player declares an action along the following lines - "I know it's a long distance, and I'm not guaranteed to make it, but I've got no choice but to give it a go!" - then I think that the commitment has been demonstrated, the risk taken (because if the dice fail, presumably the PC is falling to some degree of hurt at the bottom of the chasm) and the check called for. Why is it uncertain? Because we don't know whether, in this circumstance under this pressure this character can clear such a distance. Why is it not certain that s/he will fall in? Because we know s/he will clear some not-much-shorter distance
every time, and so there is a chance that s/he will do better than her everyday efforts at this moment of crisis.
If the player took on extra costs, like torn muscles or stopping to take a deep breath, I'd probably give a bonus (in 4e that would be +2; I'm not sure of the best way to translate that into 5e, but advantage may not be out of the question).