The linguistic processes are technically called Taboo (in this case, Taboo words) and Euphemism, which is substituting a non-taboo word for a taboo word, like saying
intead of
The way it happens is that there are always taboos on certain terms and topics in every culture. These taboo words are the healthiest words in the language, because everybody has to know them, in order to avoid saying them.
(If this sounds crazy, that's because it is -- taboos are unconscious, and not really subject to logic. After all, words by themselves have no powers; it's human culture that produces taboos.)
Anyway, people do need to talk about things, even if it's forbidden, so we substitute "safe" terms (like water closet or crapper or toilet or washroom or men's room or bathroom, instead of "place where one shits in private"). These euphemisms have a short half-life, since once the substitution strategy is detected, the euphemism gradually becomes taboo itself, and is replaced by another euphemism, while the original taboo term goes on forever.
Derogatory, politically (in)correct, profane, vulgar, racist, sexist, and other terms that are applied to language chunks are simply descriptions of the variety of taboo that the terms in question are said to be breaking. They're not categories of words so much as social infractions, which is (luckily) not a matter of grammar.
Here are the usage stats from the Corpus of Contemporary American English (number of instances per million words, by context):
SPOKEN FICTION MAGAZINE NEWSPAPER ACADEMIC
mother 281 939 287 253 134
mom 125 204 102 59 6
And now the same stats sorted by year:
1990–1994 1995–1999 2000–2004 2005–2009 2010–2012
mother 361 413 382 359 353
mom 58 94 103 122 139
While we're at it, for a more complete picture, let's also check the Corpus of Historical American English:
(X axis: year, Y axis: incidences per million words)
What we can see is that mom has indeed been gaining ground, but it's nowhere near surpassing mother.
And if you think that mom is more politically correct than mother, you must be forgetting "your mom" jokes.
Best Answer
Political correctness is in the ear of the beholder.
Personally, until I am deemed to be of the huperson race, I will continue to use man hour, manpower, mankind, etc.
And, to answer your question, don't use "person-hour" on a job-application. It sounds ridiculous (in my opinion). If you want to use something gender-neutral, you can measure in terms of FTE (Full Time Equivalent).