Leaving aside the specific piece that prompted the question, the core of what's being asked is:
How does RNA label out-of-key chords?
There are three primary ways
1. Just add a sharp or flat before the Roman numeral
Suppose we're in A minor and encounter a Bb major chord. This can be labeled bII ("flat two"). Similarly, in A major, and F major chord could be labeled bVI.
However, there is a significant caveat.
The core purpose of RNA is to describe the function of a chord. Within RNA, bII and bVI have understood meanings in terms of how those chords are expected to behave, or the contexts in which they appear. However, in terms of chord function, there's no such thing in RNA as a #III chord.
In literal terms, of course, a C# major chord occurring in the context of A minor would be #III. So if your purpose is to use Roman numerals as a generic substitute for chord symbols, then calling it #III is fine. It gets the point across. However, in functional terms, #III is meaningless and is not part of "true" RNA.
For the case of C# major in the key of A minor, we need...
2. "Secondary" chords
A secondary chord is a chord that functions in a key other than the primary one of the piece. OP proposes that a piece is in A minor, but that an F major chord could be the IV chord in the key of C major. That is to say, the F major chord functions in a key (C major) other than the primary one (A minor). Thus the F chord would be a "secondary" chord.
RNA employs a slash notation to show that a Roman numeral should be interpreted relative to another key. For example, suppose a piece in A minor modulates to C major. C major is considered the "key of III"; that is, the key based on the third scale degree of A minor. Thus, an F major chord would be IV/III: the IV chord relative to the "key of III", C major.
Let's return fully now to the piece that prompted the question. Taking at face value the OP suggestion that the piece is in A minor, then the F major chord in measure 5 and 7 is actually in key. It's the VI chord.
However, let's follow the train of thought that it is actually operating as the IV chord of C major (i.e., the "key of III"). In that case, the RNA label would be IV/III ("four of three"), meaning "the four chord relative to the 'key of III'", as explained above.
Taking this idea a step further, consider the preceding chord in measure 4. Again allowing we're in A minor, labeling this G major chord as VII makes perfect sense. But let's consider that its function is actually to lead us to the F chord in measure 5. In that case, we'd want its label to reflect its use relative to that F chord. Measure 4 would therefore be labeled II/VI ("two of six"), meaning "the two chord relative to the 'key of VI'".
Similarly, suppose measure 5 is actually the beginning of a modulation to G major. Measure 5 now becomes VII/VII, and measure 6 becomes iii/VII.
There is extensive discussion of this type of RNA notation in What is a secondary dominant chord? Although that Q&A focuses on the use of V chords, it should give a thorough understanding of the concept of "secondary chords", as these "slash chords" are known. And as shown here, the concept isn't limited only to V chords.
3. Indicate a key change and use RNA in that key
When there are a significant enough number of chords functioning in a different key — an extended modulation, for example — the new key is indicated directly and then the RNA proceeds in that key. This removes the clutter of slashes on every single chord.
This method is discussed in Use of Roman Numeral Notation when a new key is introduced within original key