In genome research, what is the problem in Mapping that may be caused by reads being too short?
Being a bioinformatician as well, I am not really what you asked for, but I work with plant genetics so I'll try and answer anyway.

What you are mapping is RNA. So, as you already figured out, splicing events will be a problem for end-to-end mapping of the reads. There are tools managing that, though, so let's assume you used one of them and still a lot of your reads do not map. To put in WYSIWIG's good point: Another event that can mess up with your alignments is RNA editing, although this is not too likely to cause a high proportion of reads to not align at all.

Some aligners might filter out "too short" queries, so make sure you are not using one of them. 

Then, have you preprocessed your reads? If you have not, there might be adaptor sequences left. Or may reads have very poor quality, thus are also aligned with poor quality, thus might be counted as unaligned.

And then check what you are aligning to. Many published plant genomes are also of minor quality, including lots of unassigned bases. So, there could be large proportions of your reference genome that count in the genome length, but are only Ns and nothing will align there.

Last but not least, your thought about a virus could also be right. Depending on the experiment, there might be pathogenic RNA in your sample, so check against a suitable database.

If the problem is only that the reads are "too short" for whatever reason, try doing transcriptome assembly before comparing to your reference.