Although a few (nutty) people continue to debate the issue, most business writers agree that it’s inappropriate to use gendered pronouns such as “he” and “his” when discussing individuals of both sexes. Instead, you can:
Rephrase the sentence. Change “The employee should bring his application to HR” to “Employees should bring their applications to HR” or “Applications should be brought to HR.”
Replace the pronoun with an article. The employee can also bring the application to HR. Similarly, “Each runner should drop the timing chip in the basket.”
Use the second person. “Bring your application to HR” and “Drop your timing chip in the basket.”
Replace the pronoun with a nounsuch as “person,” “child,” etc.
I would also stay away from awkward constructions like “s/he” and “him/her,” as well as sentences that alternate “he” and “she.” These terms distract the reader and keep the focus on gender rather than nonspecific personhood.
Pronouns aside, most words that have been deemed sexist now have gender-neutral equivalents. Examples include:
House cleaner instead of cleaning lady