Cognate

Definition and stakes

Frank Edgar Farley,  An Advanced English Grammar with Exercises

“ She sleeps the sleep of death. Note. Cognate means “kindred” or “related.” The cognate object repeats the idea of the verb, often with some modification, and may be classed as an adverbial modifier. Its difference from the direct object may be seen by contrasting “The blacksmith struck the anvil” with “The blacksmith struck a mighty blow” (cf. “struck mightily”) . For the pronoun it as cognate object, see § 120.
5. Adverbial Objective 109. A noun, or a phrase consisting of a noun and its modifiers, may be used adverbially. Such a noun is called an adverbial objective.
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Source: Gutenberg

Friedrich Kluge,  An Etymological Dictionary of the German Language (1891)

“ Although the West Teut. cognates must have existed perhaps as early as the 4th cent., nothing definite can be asserted concerning their origin and their numerous meanings, the base of which seems to be ‘to act affectionately for, or in conjunction with, some one’; to this Gr. βλέφαρον, ‘eye,’ as well as βλέπειν, ‘to see’ (Aryan root glegh?) , is perhaps primit. allied. If the cognates have been borrowed, their source cannot be determined; Rom. is out of the question, since it contains no suitable root from which they can be derived. ”
Source: Wikisource

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