MTB and Road shifters ?

Note sure if this is the right place to post but CX seems to be more appropriate for 1x questions.
I recently converted my steel road bike to run 1x9 - The front mech (band on type) was stopping me running anything bigger than 28c but with the front mech removed I can run 35c tyres which I find so much better for the dodgy country lanes around my house.
I have Shimano Sora R3000 (latest version) with a 11-32 casette
Anyway all is good however I only have the choice of 2 chain rings either a 50t or a 34t and due to the 110BCD 4 bolt crank I have limited choices for other options (unless I spend around £70 for a wolf tooth ring which I don't really want to do).
The 50t is fine for most of my rides but there are a few nice rides that I want to do that take on some uphil gravel sections and I simply do not have the legs to get up them on 50x32 and walking would take too long! The 34x32 setup obviously works fine but I then find I spin out on descents or some of the longer flat sections.
Anyway one thought I had was to use some 2nd hand MTB bits I have namely a 9 speed Deore rear mech and then swap out the 32t sprocket with a 36t sprocket - my cassette has individual sprockets so this is easy to do and a 36t sproket from Miche is only £6, The whole job including a new cable should cost me less than a tenner.
My question is would a 9 speed Shimano Deore rear mech (either medium or long cage) work with a Shimano R3000 9 speed STI lever ?
Ta.
I recently converted my steel road bike to run 1x9 - The front mech (band on type) was stopping me running anything bigger than 28c but with the front mech removed I can run 35c tyres which I find so much better for the dodgy country lanes around my house.
I have Shimano Sora R3000 (latest version) with a 11-32 casette
Anyway all is good however I only have the choice of 2 chain rings either a 50t or a 34t and due to the 110BCD 4 bolt crank I have limited choices for other options (unless I spend around £70 for a wolf tooth ring which I don't really want to do).
The 50t is fine for most of my rides but there are a few nice rides that I want to do that take on some uphil gravel sections and I simply do not have the legs to get up them on 50x32 and walking would take too long! The 34x32 setup obviously works fine but I then find I spin out on descents or some of the longer flat sections.
Anyway one thought I had was to use some 2nd hand MTB bits I have namely a 9 speed Deore rear mech and then swap out the 32t sprocket with a 36t sprocket - my cassette has individual sprockets so this is easy to do and a 36t sproket from Miche is only £6, The whole job including a new cable should cost me less than a tenner.
My question is would a 9 speed Shimano Deore rear mech (either medium or long cage) work with a Shimano R3000 9 speed STI lever ?
Ta.
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Posts
Could also go for say a 40t ring on the front and sticking with 11-32 rear.
Marin Nail Trail
Cotic Solaris
I don't want to spend more than about £20
Superstar Components Raptor 110BCD is £30 8)
So 50% over budget then.
Cannondale Synapse alloy with 'guards for the winter roads
Fuji Altamira 2.7 for the summer roads
Trek 830 Mountain Track frame turned into a gravel bike - for anywhere & everywhere
Shame I could not get a cheap narrow/wide - The superstar stuff looks decent but out of stock in 40t. Only had 1 dropped chain on the current 34t setup and that was take from a standard chainset so the 39t should be fine.
According the the gear inc calc website a 39x11 would give me a 97" gear or around 25mph at a cadence of around 85rpm so this would be ideal - Still possible to spin out on long descents but I reckon I would reach for the brakes and chicken out before my legs spin out too much!
Thanks for the advice