protective
removable, reversible
leather, plastic
dust
We spread dust covers over the furniture while the builders were working.

cushion, duvet, mattress, pillow (AmE), seat
loose (BrE)
a sofa with a loose cover

album, CD, DVD
manhole

put on, replace
lift, open, remove, take off
He held his nose as he lifted the cover of the bin.

cover for
a cover for the swimming pool

dense, thick
a thick cover of snow

cloud, ice, snow
We descended into JFK through thick cloud cover.

canopy, forest, ground, tree, vegetation, vegetative (technical, esp. AmE)
plants that provide good ground cover

back, front
hard, soft
glossy
book, magazine

adorn, grace
Her image has graced the covers of many glossy magazines.

design

illustration, image, photo, photograph, picture
art, artwork, design
article, feature, story
page
shoot (= the activity of taking a photograph for a magazine cover)
boy (AmE), girl, model (AmE)
price (= the price that a book or magazine is sold for)
The author only gets 1% of the cover price.

on a/the cover
a picture of the author on the back cover.

bed
thick, warm
thin

get under
pull up
fling off, pull back, push back, throw back
She threw back the covers and got out of bed.

pull off, rip off, yank off (esp. AmE)
‘Get up!’ she said, yanking off the covers.

tuck around sb

under the cover

pull the covers over your head
He pulled the covers over his head and tried to get back to sleep.

comprehensive, full
wide
This company provides wider cover.

standard
additional
insurance
fire, health, indemnity, life, medical

have
give sb, provide (sb with)

cover against
cover against accidental damage

cover for
cover for contents

air
The planes provided air cover for the attack.

dive for, run for, take
We ran for cover as it started to rain.

seek
find
give sb, provide (sb with)
break (= leave)
The deer broke cover as the hunters approached.

under cover
All the seats are under cover.

under cover of
We attacked at night, under cover of darkness.

cover from
They sought cover from the wind.

perfect
diplomatic, official, political
spies operating under diplomatic cover

blow (= reveal)
He realized his cover had been blown

story
The guard believed her cover story.

cover for
The club is a cover for various criminal activities.

completely, entirely
loosely
partially, partly
Her hair partially covered her face.

barely
Her dress barely covered her chest.

up
She used dried leaves and twigs to cover up the hole.

try to
She tried to cover her face with her hands.
He tried to cover his embarrassment by starting to rub his hands together.

use sth to

with
He covered the body with a cloth.

completely

in
The cars were all covered in snow.

with
The children were completely covered with mud.

barely, hardly
The payments he gets barely cover his expenses.
We've hardly covered a quarter of the course.

help (to)
be intended to
be extended to
The tax may be extended to cover books.

against
This policy should cover you against accidental injury.

for
a policy that covers you for fire and theft

Cover is used with these nouns as the subject:
ban, category, chapter, cloud, collection, correspondent, course, debris, dirt, document, drape, dust, exemption, exhibition, fee, fog, hand, index, insurance, journalist, law, magazine, mask, media, mist, patent, policy, region, report, review, rug, sand, scarf, scheme, shawl, sheet, snow, study, survey, syllabus, talk, term, veil, warranty, workshop

Cover is used with these nouns as the object:
area, array, aspect, basics, bill, blackboard, blemish, bottom, class, continent, cost, cut, desk, disappointment, distance, ear, embarrassment, essentials, expense, eye, face, fee, flank, floor, gamut, genitals, ground, head, hill, hillside, imperfection, issue, liability, mile, mileage, moon, mouth, mouthpiece, need, nervousness, news, opening, pan, period, point, possibility, postage, pot, range, rear, receiver, retreat, roof, saucepan, sidewalk, slope, span, spectrum, story, subject, surface, syllabus, territory, topic, track, trail, tuition, vent, wall, whole, wound, yard