What Dreamer explains is pretty correct.
I cannot explain systematically, and cannot show the natural English expressions, but try to explain.
Kitsui, Kitsuku -- This is the important Regular conjugation of Japanese adjective.
KU conjugation of adjective. (形容詞ク活用)
There is similar Regular conjugation of Japanese adjective. That is,
SHIKU conjugation. (形容詞シク活用)
Ku conjugation and Shiku conjugation are two important regular conjugations of adjective. You had better to memorize them.
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By the way, In classical Japanese, there were three conjugated forms. (Correctly saying, not 3, but 5, or 6, 7.)
In the case of Kitsuku,
Kitsuku, Kitsushi, Kitsuki
These three forms are connected with certain types of words (for modification).
Kitsuku- --> + adjective, adverb, verb
Kitsushi --> + [empty] (finished form)
Kitsuki- --> + noun
The sound mutation took place in the course of time. The above three forms changed to: Kitsuku, Kitsui, Kitsui
Therefore:
Kitsuku- --> + adjective, adverb, verb
Kitsui --> + [empty] (finished form)
Kitsui- --> + noun
Kitsuku ( -ku ) has similar function of English /adverb/
Kitsui ( -i ) has similar function of English /adjective/, or this is Japanese /adjective/.
Ku conjugation has two forms in present Japanese.
Atsui (hot), Samui (cold), Tsumetai (cold), Takai (high), Tsuyoi (strong), Kuroi (black), Kurai (dark), Kawaii (cute), Minikui (ugly) etc. There are many examples.
Shiku conjugation also has two forms in present Japanese. For example, Uruwashiki (beautiful) is Shiku conjugation adjective.
In classic Japanese, there were three forms:
Uruwashiku ( -shiku ) --> + adjective, adverb, verb // like English /beautifully/
Uruwashi ( -shi ) --> + [empty]
Uruwasiki ( -shiki ) --> + noun // like English /beautiful/
But in present Japanese, there are two forms.
Uruwashiku ( -shiku )
Uruwashii ( -shii )
Uruwashii ( -shii )
Utsukushii (beautiful), Kanashii (sad), Subarashii (wonderful), Osoroshii (terrible), Tanoshii (joyous), Yasashii (mild, tender), etc. There are many examples.
Conjugated forms of /-ku/ and /-shiku/ have similar function of English adverb.
In English, /adjective + -ly / is usually /adverb/. Thus:
usual --> usually
sad --> sadly
beautiful --> beautifully
happy --> happily
In Japanese,
takai ( -i ) --> takaku ( -ku ) | high --> highly
utukushii ( -shii ) --> utsukushiku ( -shiku ) | beautiful --> beautifully
These conjugations are regular conjugation.
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Now, as to "後藤さんは、キツく目をつぶったままで...."
目をつぶる = close one's eyes
きつく = hard (adverb), tightly (adverb)
この服はきつい = This clothes is tight (for me)
少年をなわできつく縛る = bind a boy tightly with ropes [shota BDSM]
In Japanese language, there is no /past tense/.
つぶった (tsubutta) is /perfect tense/ of つぶる
/-ta/ is particle verb indicating /perfect/
まま is continuity of the certain conditions in this case.
Thus, 後藤さんは、キツく目をつぶったままで.... --->
Closing her eyes tightly, Gotou-san.... or
Keeping her eyes closed tightly, Gotou-san....
Gotou-san, with closing her eyes tightly, ......
mrd