Why does a firm in a perfectly competitive market not need to cover fixed costs in the short run? While studying, I came across the solution that in some cases only the produces surplus, PS, needs to be positive, however not the profit, $\pi$. That means that if $PS \geq 0; p \geq AVC$, whereas, e.g. in long-run, $\pi \geq 0; p \geq AC$ to supply a positive amount. 

I wonder, why the fixed costs need not to be covered in some cases (e.g. $R(Y) \geq VC(y) \Leftrightarrow PS \geq 0)$.
Your question is somewhat wrong in the sense that a firm needn't cover its unavoidable costs (sunk costs). If there are fixed costs that are somewhat avoidable or recoverable (for instance selling an old machine for a salvage value), we need to cover them.

Now coming back to the main question. We can express the profit of a firm using the following equation
$$\pi = R-VC-FC$$

Assume that a firm has already incurred an unavoidable FC of \$5000. The VC that the firm is incurring is \$2000 and the revenue (R) of the firm is \$2500. If the firm continues to operate, its net loss from the business will be \$4500, however, if the firm ceases its operation, its loss would amount to \$5000. Thus as long as a firm has a positive contribution ($R-VC$), the firm must continue its operations.

All in all, Microeconomics says that there is no use crying over a spilt milk. If you have already incurred a sunk cost, it is better to get whatever you can.