Correlation matrix with a dataset with many missing value
It is impossible to say whether or not you should trim your data, without doing some Exploratory Data Analysis first. Those missing values may be important to your understanding of the dataset, so you have to look at your data comparing variable to variable to determine if the missing values hold information.
For example, if your data is about vehicles, there could be a variable representing number of doors.  If your missing data is structural an observation of a motorcycle may have a missing value for that variable.  In which case, the observation for that variable could be discarded as meaningless.  Or, you could impute the missing value to something meaningful like the number 0. It depends on what insight you are ultimately trying extract from you data.  Handling missing values is quite a complex subject.

I would suggest that you "Tidy the data" first.  From the Wickham paper


  Tidy datasets are easy to manipulate, model and visualize, and have a specific structure:
  each variable is a column, each observation is a row, and each type of observational unit
  is a table.


Initially this can help you do some Exploratory Analysis to determine what is important and what is not.  Once you have made those distinction you can find and implement strategies to handle the missing values appropriately.