If you're currently grappling a creature, and you try to shove it away (e.g. off a bridge) but you fail your Athletics check to shove it, is your grapple also broken?
When the shoving attempt fails, you don't move the creature away from you: the creature stays grappled
The Basic Rules state regarding grappling:
When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple. If you're able to make multiple attacks with the Attack action, this attack replaces one of them.
The target of your grapple must be no more than one size larger than you and must be within your reach. Using at least one free hand, you try to seize the target by making a grapple check instead of an attack roll: a Strength (Athletics) check contested by the target's Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). You succeed automatically if the target is incapacitated. If you succeed, you subject the target to the grappled condition. The condition specifies the things that end it, and you can release the target whenever you like (no action required).
The last two sentences explain what happens when you succeed, and tell us to look at the Grappled condition to see what ends it:
A grappled creature's speed becomes 0, and it can't benefit from any bonus to its speed. The condition ends if the grappler is incapacitated (see the condition). The condition also ends if an effect removes the grappled creature from the reach of the grappler or grappling effect, such as when a creature is hurled away by the thunderwave spell.
So the question is: Does the creature get moved out of reach of the character that attempts to shove it away, but fails the attempt?
Let's take a look at what the Basic Rules say about shoving a creature:
Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you. If you're able to make multiple attacks with the Attack action, this attack replaces one of them.
The target must be no more than one size larger than you and must be within your reach. Instead of making an attack roll, you make a Strength (Athletics) check contested by the target's Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). You succeed automatically if the target is incapacitated. If you succeed, you either knock the target prone or push it 5 feet away from you.
Notice my emphasis in that last sentence. It says "if it succeeds", nothing about "if it fails". So, failing to shove a grappled creature away doesn't change the distance between the grappler and the grappled.
In other words: if the attempt fails, nobody gets shoved away. And because it fails to shove it out of your reach, the creature stays grappled.
As a sidenote: You could choose to "knock someone prone" when shoving a creature. Since in that case the grappled creature also doesn't move out of reach at a successful shove, the grappled creature stays grappled regardless of the outcome of the shoving attempt.