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Every single road was covered with water that had overflowed from the river.
Literal
Road [called-という] road [topic-は] river [from-から] overflow-out water [with-で] was-covered-[resulting state-ていた].
Narrative/disaster-report register. The structure [X]という[X]は 'every single X' is an emphatic universal pattern — 'the very X called X.' 溢れだす is a compound verb 溢れる 'overflow' + 出す 'out,' intensifying to 'gush out, pour out.' 覆う is a transitive verb 'to cover'; 覆われていた is past continuous passive — 'was (being) covered / had been covered.' Vivid disaster imagery evoking flooding.