Looking for suggestions on a mostly-road mixed-use tire I'm riding tracks that are 90% on-road, 5% poorly paved cracked roads and 5% gravel, dirt, chips, etc. No mud. Basically: ride 5km to a park, ride 1km there, ride back. 

I'm pretty new to bikes. The tires I'm currently using are the stock Bontrager H5 Hard-Case Ultimate, 700x32c on my CX bike. I'm thinking that I'd like better road handling without sacrificing dirt traction.

I don't mind spending a lot on a tire. What would be some good tire choices for my use case? Which aspects would they upgrade?
Product replacements are considered off-topic, so I will discuss tire characteristic trade-offs for your riding environment:


  I'm riding tracks that are 90% on-road, 5% poorly paved cracked roads and 5% gravel, dirt, chips, etc. No mud. Basically: ride 5km to a park, ride 1km there, ride back.


Because you are not dealing with mud, a slick tire or nearly slick tire will work quite well for the short sections of gravel you are encountering.  While many people prefer knobby tires, these are not necessarily required: 


  The truth is that on gravel, knobs don’t make any difference. Without semi-firm ground to dig into, knobs can’t do anything. When you slide, it’s because gravel is sliding on gravel, not because your tires are sliding on the top layer of rocks.
  
  Why We Don’t Make “Gravel” Tires


The above is especially true for larger volume tires, example 700x40c tires at low pressure, which float on the loose gravel, providing traction through a large contact patch. (Your cross bike may or may not fit such large tires).  That said, this isn't the whole truth.  Often gravel roads can have a thin layer of loose on hard pack, here knobs will dig through the thin layer of loose to hook up on the hard packed. Slicks will not.  

Slicks however can have a surprising amount of traction on dirt, as long as you don't push the tire to the point that it slips. Kinetic friction, friction when sliding, is in general a lot lower than static friction (the friction mode tires usually operate on, you roll, you don't slide).  Furthermore, this scenario is made worse as the small rocks act as ball bearings when sliding and there slick tire has no edges to try and hook up on irregularities while sliding.  Here knobby tires will be more controlled in a slide as all the knobs provide lots of edges, and edges provide more traction when sliding. 

However your ride is only 5% gravel, a relatively small proportion of your route, so running a knobby means that 95% of your ride essentially sucks as the knobs rob speed and squirm on the pavement.  As such, the most fun will likely be had if you optimize for the 95% paved.

A higher volume slick will also work well on the poor paved roads as it lets you run lower tire pressure before 1) handling declines and 2) you bottom out the rim.  Lower pressure means a more comfortable ride in the rough road. But similar to gravel, this is a small percentage of your ride, so if doing this adds rolling resistance in the other 90% may not be ideal.  This really depends on the quality and construction of the tire (more below).

Your current tires are probably a good fit for your current ride. The only thing I would consider is that any tire with a descriptor such as "Hard-case" will be a stiff ride, which means that either comfort or speed (or both) will be sacrificed. A more supple (easier to bend) 32 tire will probably be faster. That said, unless you are running tubeless with sealant you then have to balance this out against higher probability of a puncture. (Hard-case usually implies puncture protection).

Finally, and most importantly, your rides are currently quite short (i.e., 12 km).  On shorter rides (i.e., under an hour), I am personally hard pressed to notice much difference in tires. These differences really start to get noticed when you are on longer rides (e.g., 2-6 hour rides).