If I react to my opponent's temporary trigger on their Sprite Fountain or Sprite dying at the start of their beginning phase with a Hidden Temporal Breach at a battlefield I control with a Brynhir to banish him and replay him, does my opponent still get the Sprite from the deathknell of the Sprite Fountain or are they unable to play the card?

No — your opponent still gets the Sprite.

The chain resolves in your favor for sequencing (Temporal Breach → Brynhir's "can't play cards" trigger → the Temporary kills), but Brynhir's restriction only stops opponents from playing cards. The Sprite Fountain's Deathknell is a triggered ability that plays a Sprite token, and tokens are not cards (Rule 350.2), so it triggers and resolves normally even while Brynhir's effect is active.

This matches the established ruling on Brynhir vs. Deathknell token creation (e.g., Machine Evangel, FAQ #6800): Deathknell is neither "playing a card" nor does it involve a card. Same result if you only reacted to the Sprite token's Temporary — only the Fountain's Deathknell produces the Sprite, and it isn't affected.