I wouldn �t use it in this sentence without  �the � before the noun  �hope �
 
 I �m contacting you in the hope that you can help me. - This is correct.
 
These both sound incorrect:
-> I �m contacting you in hope that you can help me. / I have to get in touch with you in hope that you can help me.
But you can say   �I �m contacting you and (I) hope that you can help me (here  �hope � is a verb)
 
I hope you can help me. (verb)
 
These are correct: 
There is no hope of finding him now.
There is little hope of finding the lost child. (= we probably won �t find the child)
The police kept searching in the hope of finding him.
 
Here is an example where both versions are correct: 
There is still hope of finding him alive.
There is still a hope of finding him aive.