The key for this question is the subjunctive mood rule.
After certain verbs and adjectives that indicate one person want another person to do something, there are two valid grammatical forms:
1) (verb/adjective) + that + subject + verb in infinitive --> (demand that HE go)
2) (verb/adjective) + objective pronoun (with verbs only) + to + verb in infinitive --> (demand him to go)
list of verbs includes: advise, command, request, ask, suggest, urge, propose, order, demand, recommend, insist, ...
list of adjectives includes: necessary, urgent, important, obligatory, imperative, mandatory, ...
Back to our question,
a) that they go back to work --> correct : note that "GO" is not following the subject verb agreement with "they", but it is in infinitive form
b) that these people go back to work --> redundant, adds ambiguity "who are these people?".
c) for them to go back to work --> adding "for" makes it incorrect.
d) that they would go back to work --> putting "would" in the middle makes it incorrect.
e) they should go back to work --> omitting "that" and putting "should" in the middle makes it incorrect.