The right ASL sign for "cover" depends on what you are covering.
The general sign for COVER:
https://youtu.be/YbIUDhz0Whc
COVER:




If you are at a restaurant and want to pay for your ASL instructor's
meal as in "I'll cover this" then the sign you want to use is "PAY."
("PAY" has several versions through so stay flexible regarding which
exact version.)
If you are covering a jar then your non-dominant hand is going to
need to be a "C" handshape (to represent the jar) and your dominant
hand can be a flat hand or a flat-palmed-claw-hand as in "LID."
If you are trying to tell someone to
"cover their mouth when they cough" you would sign HAPPEN YOU COUGH
and then mime for them the act of coughing into the crook of their
elbow. If you are talking about covering a bowl of food, you might
want to mime the act of stretching and smoothing plastic wrap onto a
bowl.
If you are discussing covering up a dead body after last
night's zombie attack then you might want to mime pulling a blanket
over the body.
If you mean cover as in "to lie in order to protect someone (or yourself) from
the consequences of telling the truth" for example "Stop covering for your
brother and tell me the truth!" -- then use the sign for "LIE
/ lying / fib"
"coverage"
At the end of 2019 I viewed a Deaf newscast in which the Deaf newscaster used
the phrase "video coverage" and included the sign COVER in the signed phrase
"VIDEO COVER"-("video coverage") while casting to a primarily Deaf audience.
As in, "I hope you have enjoyed our video coverage of this event." (The
event was the Deaflympics).
Please note that I'm not typing this to "prescribe' or "ascribe" any right or
wrongness to using the sign "COVER" to mean "coverage" -- rather I'm simply
describing the fact that the sign "COVER" did show up. It was there. It
was used. If a person viewing the newscast knew the sign he/she/they would
understand that snippet of the newscast. If a person didn't know that sign they
would experience some degree of confusion or lack of understanding of the
information.
The sign "COVER" (or - in context - "coverage") is related to but slightly
different from the sign ON. The "cover" sign tends to
involve more of a sideways "arc" onto the top of the back of the palm-down
non-dominant hand. (Verses the sign for ON which tends to involve more of a
downward movement and less of the sideways movement of the dominant hand).
_______________________
* Reference: The Daily Moth, Youtube video: QUzSUnNQtKI at 13 seconds
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