Idea used

On removing one electron, bonding becomes stronger if the electron removed is from an antibonding molecular orbital. Then bond order increases by $\tfrac{1}{2}$.

Recall: $\text{Bond order} = \frac{N_b - N_a}{2}$ where $N_b$ = number of bonding electrons, $N_a$ = number of antibonding electrons.

Check each molecule

(A) $NO$

Total electrons $= 7+8 = 15$.

In $NO$, the highest occupied MO contains one electron in a $\pi^*$ antibonding orbital.

Bond order of $NO$ is: $\text{B.O.} = 2.5$

If one electron is removed, it is removed from the antibonding $\pi^*$ orbital: $\text{B.O. increases from } 2.5 \to 3$

So bonding becomes stronger.

✅ (A) is correct.

(B) $N_2$

Total electrons $= 14$.

MO configuration up to valence shell: $\sigma 2s^2\, \sigma^{*}2s^2\, (\pi 2p_x)^2(\pi 2p_y)^2(\sigma 2p_z)^2$

Bond order: $\text{B.O.} = 3$

The highest occupied MO is $\sigma 2p_z$, which is bonding.

Removing one electron decreases bond order: $3 \to 2.5$

So bonding becomes weaker.

❌ (B) is not correct.

(C) $O_2$

Total electrons $= 16$.

MO configuration: $\sigma 2s^2\, \sigma^{*}2s^2\, \sigma 2p_z^2\, (\pi 2p_x)^2(\pi 2p_y)^2\, (\pi^{*}2p_x)^1(\pi^{*}2p_y)^1$

Bond order: $\text{B.O.} = 2$

The electron removed comes from $\pi^*$ antibonding orbital.

So: $2 \to 2.5$

Bonding becomes stronger.

✅ (C) is correct.

(D) $C_2$

Total electrons $= 12$.

MO configuration: $\sigma 2s^2\, \sigma^{*}2s^2\, (\pi 2p_x)^2(\pi 2p_y)^2$

Bond order: $\text{B.O.} = 2$

HOMO is $\pi 2p$, which is bonding.

Removing one electron decreases bond order: $2 \to 1.5$

Bonding becomes weaker.

❌ (D) is not correct.

(E) $B_2$

Total electrons $= 10$.

MO configuration: $\sigma 2s^2\, \sigma^{*}2s^2\, (\pi 2p_x)^1(\pi 2p_y)^1$

Bond order: $\text{B.O.} = 1$

HOMO is $\pi 2p$, which is bonding.

Removing one electron decreases bond order: $1 \to 0.5$

Bonding becomes weaker.

❌ (E) is not correct.

Final selection

Molecules whose bonding becomes stronger on removal of an electron: $NO \text{ and } O_2$

Therefore, the correct option is: $\boxed{\text{C: (A), (C) only}}$

Comparison with stored answer

Stored correct answer = C.

My derived answer also = C, so they agree.