Can I ask the dentist to not put numbing paste on before the novocain (procaine) shot?

Of course. Of course, you can ask the dentist anything. The numbing gel you are talking about is usually 20% lidocaine. It numbs the surface before the needle is inserted.
Yes. Of course, the topical is there to help the little pinch as the needle enters the tissue. If you dont like it let your dentist know.

Related Questions

Will I be able to request the dentist to not put the numbing paste stuff on before the novocain (procaine) shot?

Of course! There is no medical necessity for the topical anesthetic gel. Many people don't like the flavor or the temporary numbness of the throat if they swallow any of it. You can simply have the injection. That is what really numbs the tooth and/or gum for long enough to have the procedure done.
Sure. The topical anesthetic is placed since many patients do not like the needle stick. The injection can be given with or without.
Any request. You should be able to make most any request. Many patients request no anesthetic at all for small, minimally invasive dental work.