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There's nothing to do but be patient with this weather.
Literal
This weather [with-とは] patiently associate-go [only-しかない].
気長に ('patiently, calmly,' literally 'with a long spirit') is the adverbial form of 気長, contrasting with 気短 ('impatient, short-tempered'). 付き合う ('to associate with, keep company with, date') is here used metaphorically with 天気 ('weather') — 'to deal with the weather as a long-term companion.' とは marks 天気 as a comparison-target with topical/contrastive weight. The combination ~ていく + しかない gives 'have no choice but to keep X-ing.'