Step 1: Concept
Structural analysis of phosphorus oxoacids depends on identifying distinct bonding arrangements like $P=O$, $P-OH$, $P-H$, $P-O-P$, or peroxy $P-O-O-P$ linkages.
Step 2: Meaning
A peroxy linkage consists of two oxygen atoms directly bonded to one another ($-O-O-$), with each oxygen in a $-1$ oxidation state.
Step 3: Analysis
Let's evaluate the structural formulas of the options:
* **Orthophosphoric acid ($H_{3}PO_{4}$)**: Contains only $P=O$ and $P-OH$ bonds.
* **Pyrophosphoric acid ($H_{4}P_{2}O_{7}$)**: Contains a single $P-O-P$ anhydride oxygen bridge connecting the two phosphorus atoms.
* **Peroxodiphosphoric acid ($H_{4}P_{2}O_{8}$)**: As the name explicitly states, its structure contains a peroxy peroxide bridge joining the two phosphorus centers together: $(HO)_{2}O\textbf{P-O-O-P}O(OH)_{2}$.
* **Hypophosphoric acid ($H_{4}P_{2}O_{6}$)**: Features a direct phosphorus-phosphorus ($P-P$) single bond.
Step 4: Conclusion
Thus, a peroxy bond is found exclusively in peroxodiphosphoric acid.
Final Answer: (C)