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This paper considers coding for so-called partially stuck (defect) memory cells. Such memory cells can only store partial information as some of their levels cannot be used fully due to, e.g., wearout. First, we present new constructions that are able to mask $u$ partially stuck cells while correcting at the same time $t$ random errors. The process of "masking" determines a word whose entries coincide with writable levels at the (partially) stuck cells. For $u>1$ and alphabet size $q>2$, our new constructions improve upon the required redundancy of known constructions for $t=0$, and require less redundancy for masking partially stuck cells than former works required for masking fully stuck cells (which cannot store any information). Second, we show that treating some of the partially stuck cells as erroneous cells can decrease the required redundancy for some parameters. Lastly, we derive Singleton-like, sphere-packing-like, and Gilbert--Varshamov-like bounds. Numerical comparisons state that our constructions match the Gilbert--Varshamov-like bounds for several code parameters, e.g., BCH codes that contain all-one word by our first construction.