Vlan – Is it possible to access more than one VLAN

The computers (computer C and the DHCP server) will send traffic destined for the 192.168.x.0/24 blocks to the ASA, and apparently the ASA is not hair-pinning that traffic or sending ICMP redirects, which would not be terribly surprising for a firewall (or the computers aren't paying attention to the ICMP redirects).

A few options...

1) Put static routes on the DHCP server and computer C (and any other computers on that network) similar to the ones you put in the ASA.

2) Re-IP computer C and the DHCP server into a VLAN4 (or whatever) on the v1910 so that the v1910 is handling all of the internal routing...all of your endpoints can just use default routes, that point toward the v1910, and you would only have to put static routes on the ASA pointing towards the v1910. If you've got enough ports on the v1910 for your needs, you could ditch the unmanaged switch if you do this. (I like Procurves...if you've got enough ports, this or option 2a would be the way to go IMO)

2a) Re-IP the ASA in a similar way to option 2, meaning computer C and the DHCP server can keep their IP addresses.

3) There may be a way to get the ASA to issue ICMP redirects on the 172.16.3.x network so that the computers on that network can learn about the existence of the v1910. Alternatively, maybe there's a way to get the ASA to hair-pin the traffic...I don't know enough about ASA's to be able to answer that.

4) Move the v1910 to another port on the ASA, instead of daisy-chaining it off the unmanaged switch and route the traffic through the ASA as well. That would mean configuring whatever policy would be needed to get the traffic through the ASA appropriately.

If the "Switch layer 3" truly is configured for layer-3, you should not have the same subnet on each side of the switch. A host won't use layer-3 to talk to another device in the same subnet, that is a layer-2 function, and separating a layer-2 domain with layer-3 will cause you no end of problems.

Based on your diagram and comments, you have some serious design and configuration problems.

Edit:

The hosts on the VLANs should use the VLAN IP address configured on the layer-3 switch as the gateway for the host. This will allow them to connect to other hosts on different VLANs.

The hosts addressed in the subnet of a VLAN need to be connected to a port that is in that VLAN.

The layer-3 switch, as the site router, should have its default route point to the IP address of the device which is directly connected to the Internet if you wan the hosts on the VLANs to access the Internet.