Heat as a low quality energy

Energy stored as heat, by itself, is neither low- nor high-quality. What matters is the temperature at which the heat is stored, and the relationship of that temperature compared to the heat sink that will absorb the excess energy in the process.

To be more specific, say you have a heat sink at $T_S=20°:mathrm C$, such as the atmosphere for a car engine. Then the interesting comparison is between (say) $1:mathrm J$ of energy stored at $100°:mathrm C$ (such as a mass $m_{100}$ of water just below boiling point) and the same $1:mathrm J$ of energy stored in a bigger mass $m_{30}$ of water at a lower temperature of $30°:mathrm C$: although both samples have the same amount of energy, the one with a bigger temperature difference to the heat sink can operate a heat engine more efficiently and therefore can be used to perform more work (as opposed to simply handing most of its energy directly to the heat sink).

This is the reason heat is sometimes described as 'low-quality', when it is stored at low temperature (such as the heat created by friction between a car's wheels and the road) and therefore cannot be used to produce much useful work with the heat sinks we have available. Other sources of heat (like the exploding gas inside a car piston, or the glowing fuel rods in a nuclear reactor, and so on) are what you'd call 'high-quality energy' in that setting.

So the work done by the system must equal to the heat offered to the system.

All of the heat “offered” (read: added) to the system is not converted to work in a cycle. That would violate the Kelvin-Planck statement of the second law.

In order to complete a cycle some of the heat added must be rejected (discarded) by the system to the surroundings. So the net work done equals the heat added minus the heat rejected.

$$Delta U_{cycle}=Q_{net}-W_{net}=0$$ $$W_{net}=Q_{net}=Q_{added}-Q_{rejected}$$

With regard to heat being a "lower quality" energy, it may be instructive to compare it with an energy form considered "higher quality", e.g., electrical energy. I have read that the efficiency of an electric car motor is greater than 90% and can be as high as 98%. By comparison, the efficiency of an internal combustion engine varies between 30% and 45%.

Even a Carnot cycle heat engine operating in the temperature range of an automobile's internal combustion engine, between 2773 K in the combustion chamber and 300K in the atmosphere, would have a maximum theoretical Carnot efficiency of about 89%. But such an engine would operate so slowly (in order to be reversible) that the rate of work (power) would render such an engine totally impracticable. As someone once said, if you put a Carnot engine in your car you would get fantastic fuel economy, but pedestrians would be passing you by!

Hope this helps