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.