I had handed in...
This is past perfect, which is used to talk about something that occurred before some other event in the past, for example.
I had known him for some time when I first met his wife
The other event is meeting his wife, and so I use past perfect to indicate that the knowing occurred before I met his wife.
In your sentence, there is no other event, so you should not use past perfect. Simple past is fine:
I was not a perfect student when I was still studying at school...
the deadline in this case is understood to mean the deadline [for the project]. You should not make it plural, because there is only one deadline for each project.
A few minor corrections:
at school. For instance, I ...
It is not normal to start a sentence with for instance. It would be better to put a comma before it, rather than a full stop, and omit the comma after it.
studying at school
What else are you at school for? omit studying.
this is the start of a new sentence. You need a full stop, not a comma
those would refer to some shortcomings that you have already mentioned... but you have only mentioned one!
shortcomings I had
this is not idiomatic in this context. better to say "my shortcomings"
This is not idiomatic. Say either "student", or "at school". This NGram shows just how uncommon it is.
Putting that all together, you get:
I was not a perfect student when I was at school, for instance I handed in projects which were past the deadline. This was just one of my shortcomings as a student.
The last sentence is actually redundant, because "for instance" makes it clear that this is just one of many failings.