We have to account for every penny we spend on business trips
"We spend on business trips" is a noun phrase complement of "penny" so can I use relative pronoun to rewrite the sentence as follows:
We have to account for every penny which we spend on business trips
Short answer: yes.
Long answer:
Which: is used to refer to a thing mentioned in the previous clause.
Who: is used to refer to a person mentioned in the previous clause and that is the subject of the new clause.
Whom: is used to refer to a person mentioned in the previous clause but who isn't the subject of the new clause.
That: is used to refer to a thing OR a person mentioned in the previous clause.
Nothing: can replace either which, whom or that if it is not the subject of the new clause, and if it is not a non-restrictive clause.
For example, you can replace that/which in:
We have to account for every penny that/which we spend on business trips.
We have to account for every penny we spend on business trips. -> okay
But you can't here:
She healed my hand which was hurt.
She healed my hand was hurt -> wrong
Because which replaces my hand, the subject of the second clause.
Almost. Non-restrictive clauses are clauses that are introduced with a comma, and that are an optional addition to the sentence. In these, you can't replace who/which with that.
For example:
I bought a new dress, which I will be wearing to Jo's party. [non-restrictive]
I bought a new dress, that I will be wearing to Jo's party -> wrong
I bought a new dress, I will be wearing to Jo's party -> wrong
But you can turn it into a restrictive clause with the same meaning:
I was wearing the dress which I bought to wear to Jo's party. [restrictive]
I was wearing the dress that I bought to wear to Jo's party -> okay
I was wearing the dress I bought to wear to Jo's party -> okay
For more read: http://www.oxforddictionaries.com/words/that-or-which
We have to account for every penny we spend on business trips
Your sentence has a defining relative clause- we spend on business trips. It's defining because it explains which particular pennies you have to account for. Normally, you would attach it using a relative pronoun, but:
If the relative pronoun is the object of the verb then it can be omitted
In this case the object is penny, and it is the object of the verb spend. It can therefore be omitted, as in your first sentence.
If you want to include the relative pronoun, that's fine. The appropriate relative pronoun is that or which: there is no difference in meaning between these, though which tends to be preferred in more formal use.
