Please consider the following scenario
The router (Linksys E2000) has an INTERNET port where the main network is plugged in. The router has some ETHERNET/OUT ports where a switch, laptop or PC can plug in. The router has the address 192.168.X.Y
A LAN wire from one of the ETHERNET ports of the router is plugged back into the main network port.
Will main network see the router's DHCP address 192.168.X.Y?
Please let me know if I should clarify further.
Switches are transparent devices. Switches, themselves, don't broadcast anything. A switch will flood unknown unicast frames, and broadcast or (possibly) multicast frames will be sent to all switch interfaces, but switches do not originate traffic, nor do they care anything about layer-3 protocols, e.g. IP, so a switch doesn't know about IP addresses, much less broadcast them.
Each device on a network has its own addresses, and each knows what the local network is based on its configured network mask.
Based on your edit:
The Linksys router, and all consumer-grade devices, are explicitly off-topic here.
DHCP requests are broadcast from the hosts to the network, so DHCP requests will go to all switch interfaces, including the switch interface of your DHCP server (possibly running in your router chassis). The DHCP server will respond to the requesting host with the necessary information (IP address, mask, gateway address, etc.).
Although your network topology is not quite clear to me (consider including diagrams in questions) there are a few concepts to keep in mind here. The first of which is a broadcast domain.
A broadcast domain is a logical division of a computer network, in which all nodes can reach each other by broadcast at the data link layer. A broadcast domain can be within the same LAN segment or it can be bridged to other LAN segments. - Wikipedia
This essentially means that all broadcasts including DHCP are not going to make it any further than your router. If a unicast or multicast packet destined for another network reaches your router, it will be forwarded.
Another concept to consider is layer 3 switching. It is unknown to me if this applies to your network, but it is plausible that you can have a switch that does same network or inter-VLAN routing to take load off of your router and minimize packet travel. However, a switch also has a Source Address Table (SAT) used to forward traffic without flooding all switch ports with the traffic. The initial DHCP request will reach the entire broadcast domain, but then further traffic of the DHCP process will be between the server and client exclusively, and will not be seen by other devices connected to the switch or router.
I hope this answers your question. If you provide an updated explanation of your topology or a network diagram I may be able to better answer your question.
