No. SOHC = Single overheard cam , in an inline 4, it just means it has a single camshaft that’s positioned inside the head. DOHC means there are two camshafts in the head. In both instances the camshaft(s) are positioned above the combustion chamber(s).
In your applications, both engines have 16 valves, four valves per cylinder, two intake and two exhaust. There’s nothing inherent about one being better or having more power or being more economical than the other. Honda used SOHC engines quite frequently up to a few years ago. Their newer mainstream “Earth Dreams” 2.4L and 3.5L engines are DOHC though.
The Integra with the 1.8L made between 140 HP (no VTEC) for the RS trim and 195 HP(VTEC) for the very rare and very desirable Type R (if you have one, people will pay handsomely for it), the most powerful mainstream model was the GS-R with 170 HP(VTEC). If you have an automatic Integra then you have the non-VTEC 140 HP engine. Integras didn’t much much torque 120-130 lb-ft was all they had. Consequently they comparatively short gearing to make then more driver friendly in typical driving.
The HR-V is based on the Fit, it’s heavier than the Integra, But still pretty light for a compact CUV. It’s has i-VTEC, i-VTEC is different from the standard form of VTEC in that it allows for continuous adjustment of the valve timing rather than the on/off system of the standard VTEC. It makes for wider power band. The HR-V is not a fast vehicle, it will probably seem pretty pokey compared to an Integra.