Singapore Chinatown Recs
Since you are staying at Clark Quaye area, you can walk down the river to try Jumbo Seafood at Riverside Point. Jumbo serves local seafood varieties, well known for Chilli Crab (If you are willing to pay more, try the Sri Lanka crab rather than the local one), fried baby squids and mie goreng.
If you are looking for inexpensive local hawker stores, you can walk 15 minutes to Hong Lim Food Center. Unfortunately, the place is not that clean, but the price is cheap and serves many varieties of local hawker food. My two favorite stores are Outram Park Char Kway Teow #02-18 ( fried flat noodles with soya sauce, cockles, prawns etc) and Sungei Road Trishaw Laksa #02-67 (Laksa is a local version of spicy noodle soup).
Hong Lim Food Center is a very local spot. If you prefer a more touristy area, then take subway or taxi to Newton Hawker Center. If you want an air conditioned hawker center, then go to the shopping mall at Wisma Atria. There is a very clean decent food court on the 4th or 5th floor known as Food Republic. My favorite store there is the one that serves Hokkien Mie. Forgot the name of the store but you will observe a long queue there so unlikely to miss it.
Another inexpensive hawker center within 15-20 minutes walking distance from Clark Quaye is Golden Shoe Car Park (on the 2nd and 3rd floor). It caters to the office crowd in the CBD area. I am not too familiar with this area as I have not been there for the past 5 years but I think the food is slightly more expensive than Hong Lim. Also I am not sure if they opened in the weekend as the area is quiet then with most offices closed.