Can a soft shoe such as a canvas sneaker which has an arch support, encourage a little overpronation, especially if already have mild overpronation?

Not really no: Pronation occurs as a normal part of the gait cycle gait cycle. It occurs unfettered when you walk barefoot. A sneaker with an arch support if anything supports you arch. It is heath to use it in fact since it will force you to use some of your muscles that support your arch more than if you have a stiff shoe with an arch support/orthotic. A girdle has never made anyones stomach stronger ;).

If the shoe has : An arch support as you say it does it would not encourage over pronation as the arch would resist or counter the over pronation, .