It is fairly common to have the combined slop stream from the refinery processed through the Coker. That would include liquid from the flare header and drips from the fuel gas system. Both of those streams can have a lot of water in them, so it is important to remove the water before processing that stream in the Coker. As the composition of slop can vary widely, it is typically injected into the fractionator just above the shed decks so that any high boiling fractions do not contaminate the lower boiling products.
Your question seems to be more about slops management than it is about reprocessing slops in the Coker. It can be challenging to find the source of slop production. If you think it is coming from relief valves leaking by, this can be checked with an acoustic device that senses the high frequency noise generated when a relief valve is leaking. If the relief valve is leaking only during a portion of the cycle, finding this source of slop can be quite difficult. Wireless acoustic sensors could be placed on all the relief valves in an area and monitored that way over several drum cycles. That could assist in determining an intermittent source
Routine testing of the hydrocarbon phase of the slop using simulated distillation (SimDist) can show differences in the hydrocarbons produced (higher or lower boiling fractions) which can provide insight into where the slop is being generated.