I'll give you a very passionate answer here: Yes, you must display partial search results even when exact matches exist.
My background here is 10 years as a search UX consultant. My takeaway from these years is that:
- Search is unintelligent
- Data quality is never perfect
- Editorial efforts are never thorough and persistent
- Initially, and especially if search logs are missing, the designer won't know how users express themselves or all of their needs.
Unless your service is the only exception, then the search will find technically exact matches, that still does not help the user achieve her objective. If you filter out a result that would have been helpful based on a purely algorith-driven assumption, then you are jeopardizing the fundamental value of your service.
E.g. consider a search for "Antonio Carlos Jobim" on a streaming music service. You will get an exact match for the Brazilian artist. Does that mean you should exclude results for Antônio Carlos Jobim, Antônio Carlos Brasileiro de Almeida Jobim and Tom Jobim? Or should you exclude results where he is not the "artist", but rather the composer, is contributing to a compilation, is a featured artist, or is covered? In my experience, it is always best to show all results, but rank them as deftly as you can (if the content is public, a quick, cheap & good strategy is to use Google CSE). If you can afford it, provide intuitive grouping (my favourite is google.com's group results).
To further consider google.com, undisputedly the market leader of online search. Obviously it does not hide other results even if it has exact matches. Also, if it can't find enough results (often referred to by search vendors as "documents") that contain all the terms in your query, then it will show you results where one or more query terms are missing. This behaviour is consistent across all the vendors, including Yahoo, Bing, FAST and Elasticsearch. Not only has this pattern made practical sense for the vast majority of installations. It also sets user expectations.
Of course, you will find non fuzzy search services too, such as the internal searches in many apps, e.g. Apple Music. However, you will be hard pressed to find research that conclude that these search services are best of breed. Rather, such behavior is typically found in services where the search is not a key diffrentiator.
Are there no well functioning exceptions? Yes: Wikipedia. On their English-language version, note the behaviour if you type "Prometheus" into their search field: You skip the search results template entirely and are forwarded to the article on the Greek mythological figure. But upon arriving there, Wikipedia employs an additional UX safeguard: They provide a "Prometheus (disambiguation)" link that takes you to a results page tailored by a trained & active community of editors. It works well* for an organization with Wikipedia's data set and manpower, but such a persistent effort will be prohibitively expensive for most clients. The OP has not revealed neither the need nor the ability to expand into this advanced territory.
So, from a general methodology perspective, if you deviate from a universally established design pattern, you must have a very thorough, well researched rationale, or you impose a significant risk on your client.
If you anyway choose to remove non-exact matches:
- Monitor the search logs closely, and make absolutely sure that you are not filtering out results that could be useful.
- Come back here and let us know about your experience :-)
* Assumedly, based on personal experience.