Here balls up in this drawing is the definition — the National Lottery draw where the numbered balls come up. The wordplay is an anagram, flagged by "Botched", of TATTOO LIN (that's "tattoo line" with the last letter dropped for "briefly") plus NEARLY, and those fifteen letters rearrange to give NATIONAL LOTTERY. The clever bit is that "balls up" reads like another anagram indicator but is actually part of the definition.