Let's analyze each compound separately:

1. $\mathrm{H}_{4} \mathrm{P}_{2} \mathrm{O}_{7}$: This compound is also known as pyrophosphoric acid. Its structure has two phosphorus atoms (P) connected by one oxygen atom (O), forming a P-O-P bond. There are no other P-O-P bonds in its structure. Number of P-O-P bonds = 1

JEE Main 2023 (Online) 15th April Morning Shift Chemistry - Chemical Bonding & Molecular Structure Question 55 English Explanation 1

JEE Main 2023 (Online) 15th April Morning Shift Chemistry - Chemical Bonding & Molecular Structure Question 55 English Explanation 1

2. $\left(\mathrm{HPO}_{3}\right)_{3}$: This compound is also known as cyclotriphosphoric acid. It has three phosphorus atoms and three oxygen atoms, arranged in a cyclic structure with each P atom connected to two O atoms, forming a six-membered ring. In this structure, there are three P-O-P bonds. Number of P-O-P bonds = 3

JEE Main 2023 (Online) 15th April Morning Shift Chemistry - Chemical Bonding & Molecular Structure Question 55 English Explanation 2

JEE Main 2023 (Online) 15th April Morning Shift Chemistry - Chemical Bonding & Molecular Structure Question 55 English Explanation 2

3. $\mathrm{P}_{4} \mathrm{O}_{10}$: This compound is also known as phosphorus pentoxide. Its structure consists of four phosphorus atoms (P) and ten oxygen atoms (O) connected in a cyclic arrangement. In the structure, each phosphorus atom is connected to four oxygen atoms, with two terminal oxygen atoms and two bridging oxygen atoms. There are six P-O-P bonds in the P₄O₁₀ structure.

JEE Main 2023 (Online) 15th April Morning Shift Chemistry - Chemical Bonding & Molecular Structure Question 55 English Explanation 3

JEE Main 2023 (Online) 15th April Morning Shift Chemistry - Chemical Bonding & Molecular Structure Question 55 English Explanation 3

So, the number of P-O-P bonds in $\mathrm{H}_{4} \mathrm{P}_{2} \mathrm{O}_{7},\left(\mathrm{HPO}_{3}\right)_{3}$, and $\mathrm{P}_{4} \mathrm{O}_{10}$ are 1, 3, and 6, respectively.