pre-rainy-season showers; unstable weather before the rainy season
Refers to the showery, unsettled weather that often occurs in May, before the main 梅雨 (rainy season) sets in. A seasonal word (季語) for early summer.
See also: 梅雨
This year the pre-rainy-season showers have been long, and the unsettled weather continues.
During the pre-rainy-season showers, laundry doesn't dry easily, which is a problem.
走り (はしり, 'running' or 'beginning') + 梅雨 (つゆ, 'rainy season'). Literally 'running rainy season', referring to the early, intermittent rains that precede the full rainy season.