Speed vs. movement
The term speed is defined roughly here in the basic rules:
Every character and monster has a speed, which is the distance in feet that the character or monster can walk in 1 round.
For player characters, that speed is generally defined by their race, though class features may modify it or grant them other types of movement speeds (e.g. a climb speed or fly speed).
The rules on movement and position in combat state:
On your turn, you can move a distance up to your speed. You can use as much or as little of your speed as you like on your turn, following the rules here.
Your movement can include jumping, climbing, and swimming. These different modes of movement can be combined with walking, or they can constitute your entire move. However you're moving, you deduct the distance of each part of your move from your speed until it is used up or until you are done moving.
In addition to their ability to move up to their speed, characters can also take the Dash action to increase their movement for that turn:
When you take the Dash action, you gain extra movement for the current turn. The increase equals your speed, after applying any modifiers. With a speed of 30 feet, for example, you can move up to 60 feet on your turn if you dash.
Any increase or decrease to your speed changes this additional movement by the same amount. If your speed of 30 feet is reduced to 15 feet, for instance, you can move up to 30 feet this turn if you dash.
Thus, characters have a general "movement" pool on their turn that is dependent on their speed. However, this only applies to the regular movement they can use as part of their turn. Normally, characters do not have any ability to move outside of their turn.
Mantle of Inspiration is unaffected by any previous movement by the character
The Glamour bard's Mantle of Inspiration feature lets them expend a Bardic Inspiration die to give themselves and a number of other creatures temp HP. It adds:
When a creature gains these temporary hit points, it can immediately use its reaction to move up to its speed, without provoking opportunity attacks.
This ability is not affected in any way by the creature's prior movement on any turn. It simply lets them use their reaction to move immediately (as soon as they get the temp HP). The distance they can move is limited only by their speed - not by its movement on a previous turn.