Why does Ukranian "c" correspond English "h"?
heart серце
haulm солома
home сім'я
hoar сірий
horn сарна
It is not true for all words because some of them have different roots.
It looks like about Proto-Indo-European sound ḱ (palatovelar) with result of:
satemisation — it turned (in our case) to s for some branches (see red color at picture below) where Balto-Slavic (and therefore Ukrainian) is too.
Grimm's law also known as the First Germanic Sound Shift — it turned to h in Germanic (and therefore English) branch.
Just notice that it is not 100 % rule because (for example):
There can be newer word loanings or reloanings.
Another sound changings.
Specifically for your words:
haulm and солома are both from Proto-Indo-European *ḱolh₂mos;
home is from Proto-Indo-European *tḱey-, while сім'я́ “family” (not to be confused with сі́м'я “seeds”) is from Proto-Indo-European *ḱey-, however some consider them to be cognates;
hoar is both from Proto-Indo-European *(s)ḱeh₃-, while сірий “grey/gray” has several versions of etymology, however the primary one directs to the same (s)ḱeh₃-;
horn and сарна “roe deer” are both from Proto-Indo-European *ḱerh₂-.
At first I thought it to be just a coindidence, but now it looks for me as some regularity. I.e. at least some of -ḱ- in Proto-Indo-European turns into -h- in Proto-Germanic and into -s- (expressed as -с- in Cyrillic) in Proto-Slavic. However, being not a linguist, I can't say whether such a regularity was investigated and explained in any studies; maybe it makes sense to ask on Linguistics SE (in a more general way: about the evolution of Proto-Indo-European -ḱ- into Proto-Germanic -h- and Proto-Slavic -s-, at least in some of the roots).
heart
andhaulm
) are actually correct translations. Asking "why c corresponds to h" is false question. c does not correspond to h. At least not with the two random examples given.