Errrm no you can't. The monese app let's you link your avios account and "manage" it. All the info it pulls is from gate365 in terms of avios offers online and if you look at the FAQs it states..
Can I collect Avios with my Monese card?
Daniel
FollowNot at the moment.
We’re hard at work on giving you the opportunity to collect Avios through Monese – more details coming soon!