What is wrong with the claim that a cyclical heat engine does 4.00 kJ of work on an input of 24.0 kJ of heat transfer while 16.0 kJ kJ of heat transfers to the environment?

The claim is about the value of work done in the heat engine. First, you have to calculate it. The net work done by the heat engine is equal to the amount of heat transferred into the engine and out of the heat engine. Heat transfer into the system is positive, while heat transfer to the environment is negative.

$W=Q_{in}-Q_{out}$

Here, $W$ is the work done, $Q_{in}$ is the heat transfer into the engine, and$Q_{out}$ is the amount of heat transferred from the engine to the environment.

$$\begin{gathered} Q_{in}=24.0\mathrm{kJ} \\ {Q}_{out}=16\mathrm{kJ} \\ W=24.0\mathrm{kJ}-16\mathrm{kJ} \\ =8.0\mathrm{kJ} \end{gathered}$$

The work done by the engine is $8.0\mathrm{kJ}$.

According to the claim, the work done by the heat engine is$4.0\mathrm{kJ}$ . This is different from the calculated value of work done for the given amount of heat transfer that occurs in the heat engine.

So, the claim is wrong because the actual amount of work done is $8.0\mathrm{kJ}$.